Transmitting apparatus and modulation method thereof

ABSTRACT

A transmitting apparatus is disclosed. The transmitting apparatus includes an encoder to perform channel encoding with respect to bits and generate a codeword, an interleaver to interleave the codeword, and a modulator to map the interleaved codeword onto a non-uniform constellation according to a modulation scheme, and the constellation may include constellation points defined based on various tables according to the modulation scheme.

CROSS-REFERENCE TO THE RELATED APPLICATIONS

This is a continuation of U.S. application Ser. No. 17/098,679 filed Nov. 16, 2020 which issued U.S. Pat. No. 11,424,858 on Aug. 23, 2022, which is a continuation of U.S. application Ser. No. 15/725,998 filed Oct. 5, 2017, which issued U.S. Pat. No. 10,862,623 on Dec. 8, 2020, which is a continuation of U.S. application Ser. No. 14/615,148 filed Feb. 5, 2015 which issued U.S. Pat. No. 9,813,191 on Nov. 7, 2017, which claims priority from Korean Patent Application No. 10-2015-0017971, filed on Feb. 5, 2015, in the Korean Intellectual Property Office, and from U.S. Provisional Application No. 61/936,029 filed on Feb. 5, 2014 and from U.S. Provisional Application No. 61/945,868 filed on Feb. 28, 2014 in the United States Patent and Trademark Office, the disclosures of which are incorporated herein in their entirety by reference.

BACKGROUND Article I. 1. Field

Apparatuses and methods consistent with exemplary embodiments of the inventive concept relate to transmitting and receiving date using broadcasting, more particularly, to the design of non-uniform constellations used in a Bit Interleaved Coded Modulation (BICM) mapping bits at an output of an encoder and interleaver to complex constellations.

Article II. 2. Description of the Related Art

The current broadcasting systems consistent with the Digital Video Broadcasting Second Generation Terrestrial (DVB-T2) use a Bit Interleaved and Coded Modulation (BICM) chain in order to encode bits to be transmitted. The BICM chain includes a channel encoder like a Low Density Parity Check (LDPC) encoder followed by a Bit Interleaver and a Quadrature Amplitude Modulation (QAM) mapper. The role of the QAM mapper is to map different bits output from the channel encoder and interleaved using the Bit Interleaver to QAM cells. Each cell represents a complex number having real and imaginary part. The QAM mapper groups M bits into one cell. Each cell is translated into a complex number. M, which is the number of bits per cell, is equal to 2 for QPSK, 4 for 16QAM, 6 for 64QAM, and 8 for 256. It is possible to use a higher QAM size in order to increase a throughput. For example: 1K QAM is a constellation containing 1024 possible points and used to map M=10 bits. The DVB-T2 and previous standards use a uniform QAM. The uniform QAM has two important properties: possible points of constellation are rectangular, and spacing between each two successive points is uniform. The uniform QAM is very easy to map and demap.

The QAM is also easy to use since it does not need to be optimised as a function of the signal to noise ratio (SNR) or the coding rate of the channel code like the LDPC code. However, the capacity of the uniform QAM leaves a big gap from the theoretical limit, known as the Shannon limit. The performance in terms of bit error rate (BER) or frame error rate (FER) may be far from optimal.

SUMMARY

In order to reduce the gap from Shannon limit and provide a better BER/FER performance, a non-uniform constellation (NUC) is generated by relaxing the two properties of the uniform QAM, namely: the square shape and the uniform distance between constellations points.

It is an aim of certain exemplary embodiments of the present invention to address, solve and/or mitigate, at least partly, at least one of the problems and/or disadvantages associated with the related art, for example at least one of the problems and/or disadvantages described above. It is an aim of certain exemplary embodiments of the present invention to provide at least one advantage over the related art, for example at least one of the advantages described below.

The present invention is defined in the independent claims. Advantageous features are defined in the dependent claims.

Other aspects, advantages, and salient features of the invention will become apparent to those skilled in the art from the following detailed description, which, taken in conjunction with the annexed drawings, disclose exemplary embodiments of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and/or other aspects will be more apparent by describing certain exemplary embodiments with reference to the accompanying drawings, in which:

FIG. 1 is a schematic diagram of a first algorithm according to an exemplary embodiment;

FIG. 2 is a flowchart illustrating the operations of the first algorithm, according to an exemplary embodiment;

FIG. 3 illustrates the convergence of C_last with respect to one of the parameters as the first algorithm of FIGS. 1 and 2 is performed, according to an exemplary embodiment;

FIG. 4 illustrates a second algorithm according to an exemplary embodiment for determining an optimal constellation at a given SNR value S in an AWGN channel;

FIG. 5 illustrates the convergence of the constellation C_best as the second algorithm of FIG. 4 is performed, according to an exemplary embodiment;

FIG. 6 illustrates a third algorithm according to an exemplary embodiment for determining the optimal constellation at a given SNR value S in a Rician fading channel for a desired Rician factor K_rice;

FIG. 7 illustrates a fourth algorithm according to an exemplary embodiment for determining the optimal constellation at a given SNR value S in a Rayleigh fading channel;

FIG. 8 illustrates a fifth algorithm according to an exemplary embodiment for determining an optimal constellation;

FIG. 9 illustrates a process for obtaining an optimal constellation for a specific system, according to an exemplary embodiment;

FIG. 10 illustrates an exemplary BER versus SNR plot for 64-QAM using a Low-Density Parity-Check, LDPC, coding rate (CR) of 2/3 from DVB-T2 in an AWGN channel, according to an exemplary embodiment;

FIG. 11 illustrates a sixth algorithm according to an exemplary embodiment for determining an optimal constellation;

FIG. 12 further illustrates the sixth algorithm illustrated in FIG. 11 , according to an exemplary embodiment;

FIG. 13 illustrates a process for obtaining the waterfall SNR for a certain channel type according to an exemplary embodiment;

FIG. 14 schematically illustrates a process for obtaining a weighted performance measure function for an input constellation based on different transmission scenarios according to an exemplary embodiment;

FIG. 15 illustrates a process for obtaining an optimum constellation according to an exemplary embodiment;

FIGS. 16A and 16B illustrate alternative schemes for generating a candidate constellation from a previous constellation according to exemplary embodiments;

FIG. 17 illustrates a technique for reducing complexity according to an exemplary embodiment;

FIG. 18 illustrates an apparatus for implementing an algorithm according to an exemplary embodiment;

FIGS. 19 to 34 illustrate non uniform constellations according to various exemplary embodiments;

FIG. 35 is a block diagram to describe a configuration of a transmitting apparatus according to an exemplary embodiment;

FIG. 36 is a block diagram to describe a configuration of a receiving apparatus according to an exemplary embodiment; and

FIG. 37 is a flowchart to describe a modulation method according to an exemplary embodiment.

DETAILED DESCRIPTION OF THE EXEMPLARY EMBODIMENTS

Various exemplary embodiments will now be described in greater detail with reference to the accompanying drawings.

In the following description, same drawing reference numerals are used for the same elements even in different drawings. The matters defined in the description, such as detailed construction and elements, are provided to assist in a comprehensive understanding of the invention. Thus, it is apparent that the exemplary embodiments can be carried out without those specifically defined matters. Also, well-known functions or constructions are not described in detail since they would obscure the exemplary embodiments with unnecessary detail.

The following description of the exemplary embodiments with reference to the accompanying drawings is provided to assist in a comprehensive understanding of the inventive concept, as defined by the claims. The description includes various specific details to assist in that understanding but these are to be regarded as merely exemplary. Accordingly, those of ordinary skill in the art will recognize that various changes and modifications of the embodiments described herein can be made without departing from the scope of the inventive concept.

The same or similar components may be designated by the same or similar reference numerals, although they may be illustrated in different drawings.

Detailed descriptions of techniques, structures, constructions, functions or processes known in the art may be omitted for clarity and conciseness, and to avoid obscuring the subject matter of the exemplary embodiments.

The terms and words used herein are not limited to the bibliographical or standard meanings, but, are merely used by the inventors to enable a clear and consistent understanding of the exemplary embodiments.

Throughout the description and claims of this specification, the words “comprise”, “contain” and “include”, and variations thereof, for example “comprising”, “containing” and “including”, means “including but not limited to”, and is not intended to (and does not) exclude other features, elements, components, integers, steps, operations, processes, functions, characteristics, and the like.

Throughout the description and claims of this specification, the singular form, for example “a”, “an” and “the”, encompasses the plural unless the context otherwise requires. For example, reference to “an object” includes reference to one or more of such objects.

Throughout the description and claims of this specification, language in the general form of “X for Y” (where Y is some action, process, function, activity or step and X is some means for carrying out that action, process, function, activity or step) encompasses means X adapted, configured or arranged specifically, but not necessarily exclusively, to do Y.

Features, elements, components, integers, steps, operations, processes, functions, characteristics, and the like, described in conjunction with a particular aspect, embodiment, example or claim of the inventive concept are to be understood to be applicable to any other aspect, embodiment, example or claim described herein unless incompatible therewith.

The exemplary embodiments may be implemented in the form of any suitable method, system and/or apparatus for use in digital broadcasting, for example in the form of a mobile/portable terminal (e.g. mobile telephone), hand-held device, personal computer, digital television and/or digital radio broadcast transmitter and/or receiver apparatus, set-top-box, etc. Any such system and/or apparatus may be compatible with any suitable existing or future digital broadcast system and/or standard, for example one or more of the digital broadcasting systems and/or standards referred to herein.

A non-uniform constellation (NUC) according to an exemplary embodiment may be generated or obtained using any suitable method or algorithm including steps (or operations) for generating or obtaining such a non-uniform constellation. The non-uniform constellation according to the embodiment may be generated or obtained by any suitably arranged apparatus or system including means for generating or obtaining such a non-uniform constellation. The methods or algorithms described herein may be implemented in any suitably arranged apparatus or system including means for carrying out the method or algorithm steps.

Certain exemplary embodiments provide an algorithm for obtaining a non-uniform constellation. The non-uniform constellation obtained in the certain exemplary embodiments may provide a higher capacity than an equivalent uniform constellation (e.g. a uniform constellation of the same order). Certain exemplary embodiments may obtain an optimised non-uniform constellation using an algorithm with relatively low complexity and relatively high computational efficiency. For example, an algorithm in certain exemplary embodiments may obtain an optimised non-uniform constellation much faster than an algorithm using a brute force method that searches all (or a high proportion of) possible candidate constellations. Certain exemplary embodiments provide an algorithm for obtaining optimised non-uniform constellations suitable for very high-order constellation (e.g. having more than 1024 constellation points).

Various embodiments are described below in which non-uniform (NU) Quadrature Amplitude Modulation (QAM) constellations are obtained. However, the skilled person will appreciate that the inventive concept is not limited to QAM constellations, but may be applied to other types of constellation.

As mentioned above, a constellation may be characterised by a number of parameters, for example specifying spacings between constellation points, or specifying the position of each positive real level (complete constellations may be obtained from these parameters because the constellations are the same for real and imaginary axes and the same for positive and negative values). In order to obtain an optimum constellation, a brute force approach may be taken in which combinations of values for each of the parameters are searched with a certain step size up to a certain maximum value. Each combination of values for each parameter corresponds to a distinct constellation. The constellation having the best performance is selected.

However, in certain exemplary embodiments, the number of parameters may be reduced by imposing one or more certain geometric and/or symmetry constraints on the constellations. For example, one constraint may be that the constellations are symmetric among the four quadrants of the constellations. In addition, the constellations may be constrained in that the constellation points are arranged in a QAM type lattice in which, within each quadrant, (i) constellation points are arranged in horizontal and vertical lines, (ii) the number of horizontal lines is the same as the number of vertical lines, (iii) the same number of constellation points are arranged in each horizontal line, and (iv) the same number of constellation points are arranged in each vertical line. In another example, a constellation may be constrained to be a circular constellation (e.g. a constellation having circular symmetry). Furthermore, constellations having the same relative arrangement, differing only in size, may be regarded as equivalent. In this case, one of the parameters may be set to a fixed value. The skilled person will appreciate that the inventive concept is not limited to the above examples, and that one or more additional or alternative constraints may be used.

In certain exemplary embodiments, a non-uniform QAM (NU-QAM) constellation may have a constellation conforming to one or more geometric and/or symmetry constraints, for example one or more, or all, of the above constraints, or a rotation and/or scaling thereof. A non-uniform N-QAM constellation may be a non-uniform QAM constellation including N constellation points.

By applying the constraints described above, the number of parameters may be reduced, for example to 1, 3, 7, 15, 31 and 63 parameter(s) for constellations including 16, 64, 256, 1024, 4096 and 16384 constellation points, respectively. The number of parameters in a reduced set of parameters may be denoted by b. For example b=1 for 16-QAM (in which there are 16 positions that are symmetric on the real/imaginary and positive/negative axes). Thus there are only 2 points to define. Since the total power of the constellation is typically normalized to one, fixing one parameter will fix the other. Thus b=1 for a square 16QAM.

In certain exemplary embodiments, combinations of values for each of the b parameters are searched with a step size d up to a maximum value A. Thus, the number of search iterations is equal to (A/d)^(b).

A first algorithm according to certain exemplary embodiments for obtaining an optimum non-uniform constellation for a given SNR will now be described. The algorithm uses an iterative scheme to gradually modify an initial constellation until the constellation converges. For example, the initial constellation may be a uniform constellation, the constellation may be modified by changing the values of the parameters between iterations, and convergence occurs when the values of all the parameters change by less than a threshold amount between iterations. An optimum constellation may be defined as the constellation having the best performance according to any suitable measure. For example, the measure may include a coded modulation (CM) capacity or BICM capacity. In the following example a non-uniform 64-QAM constellation is obtained, in which the (reduced) number of variable parameters, b, is equal to 3.

FIG. 1 is a schematic diagram of a first algorithm, according to an exemplary embodiment, and FIG. 2 is a flowchart illustrating operations of the first algorithm, according to an exemplary embodiment. In the algorithm, the following variables are used. A parameter C_last denotes a particular constellation, corresponding to a particular set of values of the b parameters. A parameter C_last is initialised with a certain initial constellation, for example a uniform constellation. A parameter SNR denotes a signal-to-noise ratio. The SNR parameter is set to a desired value equal to an SNR for which an optimum constellation is desired. The parameter C_best denotes a constellation that maximises performance, for example maximises the CM capacity or BICM capacity, for a given SNR. The parameter d denotes a first step size used in the algorithm. The parameter d (or step) is initialised to a suitable value that may be determined theoretically and/or experimentally. A parameter Min_Step denotes a minimum allowed value for d, and is set to a fixed value.

In operation 201, C_last is initialised to an input constellation. In a next operation 203, step d is initialised to a value Ini_step. In operation 205, a set of candidate constellations is obtained. The set of candidate constellations includes the constellation C_last and one or more modified constellations, where each modified constellation is obtained by modifying one or more of the parameter values defining C_last using any suitable scheme. In the illustrated example, the set of candidate constellations are created based on C_last and step size d, denoted by function CreateSet(C_Last, d). For example, for each constellation point, three derived constellations are generated [C_last, C_last+d, C_last−d]. Specifically, a set of constellations is derived such that the values of the b parameters in C_last are each set to one of n new values varying around the current parameter value. For example, three new values (n=3) may be used, which include (i) the current parameter value, (ii) a value d greater than the current parameter value, and (iii) a value d less than the current parameter value. For example, if there are two constellation levels to be defined then the number of combinations to be tested are 3×3 (corresponding to three positions for each level). All combinations of the new parameter values are used to generate the set of constellation. Thus, the set of constellations includes a total of n^(b) constellations. Although three new values for each parameter are used in the embodiment described above, any suitable number of new values may be used in another embodiment. The set of new values may include the old (or current) value, or may not include the old value.

In certain exemplary embodiments, three values of each level are chosen so that the total number of possibilities to be tested is 3^(b), where b is the number of levels (parameters) to be optimised. In the case of very high-order constellations, for example above 1K, 3^(b) may be very high. In this case, all the levels may be fixed except one, for which three possibilities are tested, C_last, C_last+d and C_last−d until convergence is achieved. The same operation may then be repeated for the other levels. The cost of this operation is multiplicative and not exponential (for example, if it is supposed that each level converges in one iteration then the cost will be 3×b instead 3^(b).)

In operation 207, the performance of each constellation in the set of derived (candidate) constellations is calculated or determined using any suitable performance measure (e.g. capacity). In operation 209, the candidate constellation having the best performance (e.g. the candidate constellation that maximises the capacity) is assigned to C_best. In operation 211, it is determined whether C_best differs from C_last by more than a threshold amount. For example, in the illustrated example, the threshold amount is equal to zero, so that it is determined whether C_best=C_last. That is, it is determined whether there is any difference between constellation C_best and constellation C_last (e.g. within a certain resolution). The difference may be any suitable measure of difference, for example including a difference based on geometry (e.g. differences in the locations of the constellation points of the constellations) and/or a performance measure (e.g. a difference in a certain performance measure between the constellations). If it is determined in operation 211 that C_best C_last, then in operation 213, C_last takes the value C_best (i.e. so that the value of C_last in the next iteration is equal to the value of C_best in the current iteration) and the method returns to operation 205 in which a set of candidate constellations are created based on C_last and step size d, CreateSet(C_Last, d). On the other hand, if it is determined in operation 211 that C_best=C_last, then, in operation 215, C_last takes the value C_best and the method moves to operation 217.

In operation 217, it is determined whether d<Min_Step. If it is determined in operation 217 that d≥Min_Step then the method moves to operation 219 in which the step size d is reduced. For example, d is divided by a certain factor (e.g. 2). Following operation 219, the method returns to operation 205 in which a set of candidate constellations are created based on C_last and step size d (i.e., reduced d), CreateSet(C_Last, d). On the other hand, If it is determined in operation 217 that d<Min_Step then the value of C_best is saved and the algorithm ends.

FIG. 3 illustrates the convergence of C_last with respect to one of the parameters as the first algorithm of FIGS. 1 and 2 is performed. Initially, the value of the parameter converges to a certain value. When the value of the parameter has converged within a certain resolution, the step size d is reduced and the value of the parameter converges further, until the step size d has reached the minimum step size.

In the example shown in FIG. 3 , for each iteration, three new parameter values are tried, as represented by the vertical columns of circles. The best new parameter for each iteration is indicated in FIG. 3 as a filled circle. The best parameter value in one iteration is used as the new parameter value for the next iteration. Thus, in the example illustrated in FIG. 3 , in which three new parameter values are tried (including the current parameter and parameters an amount d above and below the current parameter), the filled circle of one iteration corresponds to the middle of the three circles arranged in a column for the next iteration.

In certain exemplary embodiments, operations 217 and 219 of the algorithm illustrated in FIG. 2 may be omitted so that operations 205, 207, 209, 211, 213 and 215 are performed using the initial step size. In this case, when it is determined in operation 215 that C_best=C_last, the step size is not reduced, but rather the value of C_best is saved and the algorithm ends. By omitting operations 217 and 219, the algorithm may potentially complete more quickly. However, in this case the output constellation C_best may differ from the true optimum constellation more than the output constellation C_best obtained in the algorithm illustrated in FIG. 2 where the step size d is decreased. This may be seen in FIG. 3 , where it can be seen that the best parameter value in the final iteration lies closer to the optimal value (indicated by the horizontal line) than the best parameter value at the stage of convergence with the initial step size.

The first algorithm described above determines an optimum constellation based on a certain performance measure (e.g. capacity). In the following, various algorithms for determining an optimum constellation for a transmission system defined by a set of one or more system parameter values, where the constellation is optimised for a certain desired value of a system parameter (e.g. a certain SNR value or certain Ricean factor). In these embodiments, a system parameter value is set to an initial value (e.g. a relatively high value) and an optimum constellation is generated using an algorithm described above (e.g. the algorithm illustrated in FIG. 2 ), wherein the performance measure is based on a defined transmission system having the set system parameter value. The system parameter value is then reset to a modified value (e.g. by reducing the value by a certain step size) and the algorithm is re-run. The other system parameter values may remain fixed. This process is repeated until the system parameter value reaches a certain desired value.

For example, FIG. 4 illustrates a second algorithm for determining an optimal constellation at a given SNR value S in an additive white Gaussian noise (AWGN) channel. In operation 401, the algorithm is initialised by setting an SNR parameter to a high value N, where N is large. For example, the initial SNR value may be set to an SNR value above which a non-uniform constellation provides no better performance than an equivalent uniform constellation. This value may be determined, for example, theoretically and/or experimentally. In operation 401, the parameter C_last is also initialised to a certain constellation, for example a uniform constellation.

In operation 403, the first algorithm described above is run using the initialised constellation C_last as the input constellation and using the initialised SNR ratio. By applying the first algorithm, the constellation C_last will converge to an optimal constellation C_best for a specific input value of SNR. An output of operation 403 is C_best obtained using the first algorithm. In operation 405 the SNR value is reduced by a certain amount, for example one unit or step size. In operation 405, C_last takes the value of C_best (i.e. so that the value of C_last in the next iteration is equal to the value of C_best in the current iteration). In operation 407 it is determined whether SNR<S. If it is determined in operation 407 that SNR≥S, then the method returns to operation 403, in which the first algorithm is run with new values of C_last and SNR. On the other hand, if it is determined in operation 407 that SNR<S, then the value of C_best is saved and the algorithm ends. By applying the second algorithm, the resulting constellation C_best is the optimal constellation for the desired SNR value S.

FIG. 5 illustrates convergence of the constellation C_best according to the second algorithm of FIG. 4 is performed. Each of the three curves represents variation in the value of a respective one of the three variable parameters. The solid constant line represents the fixed value of a fixed parameter. As shown in FIG. 5 , at the start of the second algorithm, starting from the right-hand side of FIG. 5 , the SNR value is high and the constellation is a uniform constellation, as defined by the values of the parameters on the right-hand side of FIG. 5 , labelled “Initial condition”. At each iteration, an optimal constellation is obtained for the specific SNR value (indicated in FIG. 5 by the markers). The SNR is then reduced and the optimal constellation is obtained for the new SNR (this process being indicated for one of the parameters by the stepped line in FIG. 5 ). As shown in FIG. 5 , the values of the parameters corresponding to the optimal constellation vary smoothly with varying SNR values. The iterations are repeated until the SNR value reaches the desired SNR value S.

By running the second algorithm illustrated in FIG. 4 , an optimal constellation is derived from each of a set of SNR values. These constellations may be stored in association with the corresponding SNR values, for example in a look-up table.

FIG. 6 illustrates a third algorithm for determining an optimal constellation at a given SNR value S in a Rician fading channel for a desired Rician factor K_rice. The Rician channel is given by:

$\begin{matrix} {\sqrt{\frac{K}{K + 1}} + {\sqrt{\frac{1}{K + 1}}h}} & \left\lbrack {{Equation}1} \right\rbrack \end{matrix}$

In Equation 1, K is the Rician factor and h is Rayleigh distributed (centred and normalised). Initially, the third algorithm applies the second algorithm described above to obtain the optimal constellation C_best at an SNR value S for an AWGN channel, C_best(AWGN). In operation 601, parameter C_last is initialised to C_best(AWGN). In operation 601, the Rician factor K is initialised to a high value, which may be determined theoretically and/or experimentally. For example, K may be initialised to a value K_rice+N, where N is large.

In operation 603, the first algorithm described above is run using the initialised constellation C_last as the input constellation and using the initialised Rician factor K to obtain an optimal constellation C_best. In operation 605, the Rician factor K is reduced by a certain amount, for example by one unit. In operation 605, C_last takes the value of C_best (i.e. so that the value of C_last in the next iteration is equal to the value of C_best in the current iteration). In operation 607 it is determined whether K<K_rice. If it is determined in operation 607 that K≥K_rice, then the method returns to operation 603, in which the first algorithm is run with new values of C_last and K. On the other hand, if it is determined in operation 607 that K<K_rice, then the value of C_best is saved and the algorithm ends. By applying the third algorithm, the resulting constellation C_best is the optimal constellation for the desired Rician factor K_rice.

FIG. 7 illustrates a fourth algorithm for determining an optimal constellation at a given SNR value S in a Rayleigh fading channel. A Rayleigh fading channel is a special case of Rician fading with the Rician factor K=0. Accordingly, the fourth algorithm is the same as the third algorithm described above, except that K_rice is set to zero.

Table 1 below compares the number of capacity calculation function calls for obtaining optimal constellations for various constellation sizes (16-QAM, 64-QAM and 256-QAM) using an exhaustive search, a restricted exhaustive search and an algorithm according to the present embodiment. The values in Table 1 are based on a step size d of 0.0125 and maximum value for the parameters of 10. Table 1 also indicates a factor difference between using a restricted exhaustive search and a search using an algorithm according to the present embodiment. As can be seen, the algorithm according to the present embodiment is significantly more efficient, for example by a factor of 1.15×10¹⁰ for 256-QAM.

TABLE 1 Algorithm Restricted according to Exhaustive exhaustive exemplary Gain versus search search embodiment restricted  16QAM 800 800   21   38  64QAM  5.1e9  1.9e8  1701 117577 256QAM  2.1e21  2.5e15 216513    1.15e10

In Table 1, the difference between the exhaustive search and the restrictive exhaustive search is as following. It is assumed in the following that there are 4 levels (parameters) between 0 and 10. In the exhaustive search, each of the 4 parameters is searched over the whole range [0-10] with a certain granularity. In the case of the restricted exhaustive search, the range in which each level will fall is fixed. For example level 1 (first parameter) will be in the range [0-2.5], level 2 in the range [2.5-5], level 3 in the range [5-7.5], level 4 in the range [7.5-10]. In this manner, the number of possibilities is reduced.

FIG. 8 illustrates a fifth algorithm for determining an optimal constellation. This algorithm corresponds closely to the algorithm illustrated in FIG. 2 , but is modified to increase overall efficiency. This algorithm includes an inner loop for operations (operations 803-819) corresponding to operations 203-219 of FIG. 2 . However, operation 805 for creating a set of candidate constellations is modified from the corresponding operation 205 of FIG. 2 . Specifically, in the algorithm of FIG. 8 , rather than modifying each of the b parameters and trying all combinations of the new parameters as in the algorithm of FIG. 2 , only one parameter is modified at a time. For example, within one iteration of the inner loop 803-819, only one parameter (parameter i) is modified to produce a set of candidate constellation. The capacities of these constellations are calculated and the best constellation is selected, as in FIG. 2 .

In the algorithm of FIG. 8 , the value of i is varied from 1 to b using an outer loop (operations 821-825). The algorithm of FIG. 8 is initialised in operation 801, corresponding to operation 201 of FIG. 2 . It can be seen that, by using the algorithm of FIG. 8 , rather than the algorithm of FIG. 2 , the total number of candidate constellation tried (i.e. the total number of capacity calculations) is significantly reduced. However, in simulations, the optimal constellation obtained using the algorithm of FIG. 8 is very close to the optimal constellation obtained using the algorithm of FIG. 2 , which in turn is very close to the true optimal constellation obtained using an exhaustive search. The improvement in computational efficiency using the algorithms according to the above embodiments, when compared to an exhaustive search, increases as the constellation order increases.

As with the algorithm illustrated in FIG. 2 , in certain exemplary embodiments, operations 817 and 819 of the algorithm illustrated in FIG. 8 may be omitted.

According to the above embodiments, optimal constellations may be obtained for particular parameters, for example SNR, Rician factor etc. These optimum constellations are obtained independently of any particular system implementation, for example independent of a particular coding scheme. In the following, various embodiments are described for obtaining an optimal constellation for a specific transmission system.

A transmission system may include a number of processes which may affect the optimal constellation, for example FEC encoding, bit interleaving, demultiplexing bits to cells, mapping cells to constellations, cell interleaving, constellation rotation, in-phase and quadrature phase (I/Q) component interleaving, inter-frame convolution and inter-frame block interleaving, and multiple-inputs-single-output (MISO) precoding. A QAM mapper is used in the Bit Interleaved Coded Modulation (BICM) chain to map bits to symbols. The QAM mapper may use a uniform constellation to map bits to cells (for example as done in DVB-T2). However, an increase in capacity may be achieved by using a fixed non-uniform constellation. A non-fixed non-uniform constellation (e.g. QAM) may be used to further increase capacity. The BICM capacity depends on the bit to cell mapping used. Optimisations are desirable in the LDPC design, the QAM mapping and the mapping of bits to cells.

In certain exemplary embodiments, different constellations are generated using a certain step size. A bit error rate (BER), a block error rate and/or a packet error rate corresponding to the constellations are obtained, and the best constellation is selected based on one or more of the aforementioned error rates.

In certain exemplary embodiments, the process illustrated in FIG. 9 may be carried out to obtain an optimal constellation for a specific system. In operation 901, a uniform constellation (e.g. uniform QAM) is selected. In operation 903, BER values for the selected uniform constellation are obtained over a range of SNR values (e.g. using simulation or by obtaining the BER values theoretically or experimentally). These values may be obtained based on a specific system, for example using a particular coding scheme (e.g. LDPC coding with a certain parity check matrix) with a certain coding rate and a certain bit interleaver and cell interleaver. FIG. 10 illustrates an exemplary plot for 64-QAM using an LDPC coding rate of 2/3 from DVB-T2 in an AWGN channel.

In operation 905, an SNR at which the BER falls below a threshold value (e.g. 0.001) is determined. The threshold value may be selected such that the resulting SNR falls within a “waterfall zone” of the BER curve (i.e. the zone at which the BER falls relatively rapidly with increasing SNR). The determined SNR value may be denoted S and referred to as a “waterfall” SNR.

Next, an optimal constellation may be obtained for the SNR value S determined in operation 905.

For example, in some exemplary embodiments, in operation 907 a, an optimal constellation may be selected from optimal constellations obtained when performing the algorithms described above in relation to FIGS. 1-8 (and stored in a look-up table). Specifically, an optimal constellation previously determined for the SNR value S may be retrieved from the look-up table.

Alternatively, an iterative process may be performed to obtain an optimal (non-uniform) constellation. Specifically, after operation 905, the process moves to operation 907 b in which the algorithms described above in relation to FIGS. 1-8 are used to obtain an optimal constellation for the SNR value S (or for a value close to S). After operation 907 b, the process returns to operation 903, in which BER values are obtained over a range of SNR values. In this iteration, the BER values are obtained for the optimal constellation obtained in operation 907 b (rather than for the initial uniform constellation as in the first iteration). In a similar manner as previously described, the SNR value at which the BER falls below a threshold value (using the new set of BER values for the optimal constellation) is determined in operation 905, and a new optimal constellation for the newly determined SNR value is obtain in operation 907 b. The previously described operations 903, 905, 907 may be repeated a certain number of time (for example a predetermined number of times). Alternatively, the algorithm may terminate when the waterfall SNR stops decreasing between iterations, and instead starts increasing.

FIGS. 11 and 12 illustrate a sixth algorithm for determining an optimal constellation. This algorithm corresponds closely to the algorithm illustrated in FIG. 8 , but is modified to improve performance. In particular, this algorithm introduces a concept of a direction of convergence of a parameter value. For example, within the inner loop of the algorithm, the direction is initialised to 0. When creating a set of candidate constellations, the candidate set depends on the direction parameter. When the best constellation is selected in operation 1109, the direction of convergence of the value of parameter i is obtained. For example, if the parameter value is converging upwards, then the direction parameter may be set to +1, if the parameter is converging downwards, then the direction parameter may be set to −1, and if the parameter does not change, then the direction parameter may be set to 0. As illustrated in FIG. 12 , the number of candidate constellations may be reduced when the parameter value is converging upwards or downwards.

As described above, an optimum constellation may be obtained for a particular system implementation, and/or for certain system parameter values. For example, an optimum constellation (e.g. a constellation that optimises the BICM capacity) may be obtained for a certain propagation channel type (e.g. AWGN, Rayleigh or Typical Urban, TU6, channel) and for a certain SNR. However, in some cases, data may be transmitted in different scenarios. For example, data may be transmitted through different types of channels and may be received with different SNRs. Furthermore, it may be desirable or required that a data transmission system uses the same constellation, regardless of the scenario (e.g. channel type or SNR), for example in order to reduce system complexity. In some cases, a transmission system may use a certain constellation for many different scenarios (e.g. channel types and SNRs).

FIGS. 13-16 illustrate an algorithm for obtaining a constellation that is optimised (e.g. achieves the best capacity) with respect to two or more different scenarios (e.g. different channel types and/or SNR values). The algorithm includes a number of different parts. First, the waterfall SNR for each channel type (e.g. propagation channel type) is obtained using an algorithm similar to the algorithm illustrated in FIG. 9 . A weighted performance measure function (e.g. weighted capacity) for an input constellation is defined, based on different scenarios (e.g. different channel types and SNR values). Then, an algorithm similar to the algorithms illustrated in FIG. 2, 8 or 11 is applied to determine an optimum constellation, where the performance measure is used based on the weighted performance measure.

FIG. 13 illustrates a process for obtaining the waterfall SNR for each channel type. Each channel type is treated separately in order to obtain its waterfall SNR. In particular, the process illustrated in FIG. 13 is repeated for each channel type to obtain a respective waterfall SNR for that channel type. The process illustrated in FIG. 13 operates in substantially the same manner as the algorithm illustrated in FIG. 9 , and therefore a detailed description will be omitted for conciseness. However, rather than outputting an optimal constellation, as in the algorithm illustrated in FIG. 9 , the process illustrated in FIG. 13 instead outputs the waterfall SNR determined in the final iteration of the process. The process illustrated in FIG. 13 (including BER simulation and capacity optimisation operations) is performed based on a certain channel type, and the output waterfall SNR is determined as the waterfall SNR associated with that channel type.

FIG. 14 schematically illustrates a process for obtaining a weighted performance measure function for an input constellation based on different transmission scenarios. In this example, the weighted performance measure is a weighted capacity, and the different scenarios include different channel types and associated waterfall SNR values. As illustrated in FIG. 14 , a candidate constellation is provided as an input. For each channel type and associated waterfall SNR, the BICM capacity for the input constellation based on the channel type and waterfall SNR is obtained. Each obtained BICM capacity is then multiplied by a respective weight and the weighted BICM capacities are added together to obtain an output weighted average BICM capacity. The weights may be selected according to any suitable criteria. For example, a relatively common or important channel type may be associated with a relatively large weight.

FIG. 15 illustrates a process for obtaining an optimum constellation. The process illustrated in FIG. 15 operates in substantially the same manner as the algorithm illustrated in FIG. 2, 8 or 11 , and therefore a detailed description will be omitted for conciseness. However, when determining the performance of a candidate constellation in the process illustrated in FIG. 15 , the performance is determined based on the weighted performance measure described above in relation to FIG. 14 .

In the process illustrated in FIG. 15 , in some situation, a certain constellation may achieve the best performance with respect to the weighted performance measure, even though the performance of that constellation with respect to the BICM capacity based on an individual channel and SNR may be relatively low. In certain exemplary embodiments, to ensure that a constellation obtained using the algorithm is able to achieve at least a certain level of performance for one or more, or all, transmission scenarios, an additional criterion may be applied when testing each candidate constellation to obtain the constellation C_best. Specifically, any candidate constellation that does not achieve at least a threshold performance with respect to one or more certain individual scenarios, or all scenarios, is ignored and cannot be selected as C_best, even if that constellation achieves the best performance with respect to the weighted performance measure.

In the process illustrated in FIG. 15 , a set of candidate constellations may be derived using any suitable method, for example the method described above in relation to FIG. 9 based on a step size d. FIGS. 16A and 16B illustrate alternative schemes for generating a candidate constellation from a previous constellation, C_last, that may be used in certain exemplary embodiments. In FIGS. 16A and 16B, the open circles represent the constellation points of a previous constellation, C_last. For each constellation point of the previous constellation, a respective set of N modified constellation points are defined, as indicated in FIGS. 16A and 16B as filled circles. Each set of modified constellation points forms a pattern of constellation points located relatively close to the respective constellation point of the previous constellation.

For example, as illustrated in FIG. 16A, each set of modified constellation points may form a square or rectangular lattice of N=8 constellation points surrounding a respective constellation point of the previous constellation. The lattice spacing is equal to d. Alternatively, as illustrated in FIG. 16B, each set of modified constellation points may form a ring of N=8 constellation points surrounding a respective constellation point of the previous constellation. The radius of the ring is equal to d.

A candidate constellation may be obtained by selecting, for each constellation point in the previous constellation, either the constellation point of the previous constellation itself or one of the constellation points of a respective set of modified constellation points.

In the examples described above, a weighted performance measure is defined based on different transmission scenarios. For example, in the case illustrated in FIG. 14 , each transmission scenario includes a different channel type and an associated waterfall SNR value. Accordingly, a constellation optimised for a range of channel types and associated SNR values may be obtained. In an alternative embodiment, an optimal constellation may be obtained for different transmission scenarios, in the case where each transmission scenario includes the same channel type, but involves different SNR values (e.g. a set of SNR values S1, S1+d, S1+2d, S1+3d, . . . , S2, where d is a step size). That is, an optimal constellation may be obtained for a fixed channel type that is intended to be used over a range of SNR values. In this case, the algorithm described above in relation to FIGS. 13-16 may be used, except that when determining the weighted performance measure as illustrated in FIG. 14 , instead of determining individual BICM capacities based on respective channel types and associated waterfall SNR values, the individual BICM capacities are determined based on the fixed channel type and respective SNR values S1, S1+d, S1+2d, S1+3d, . . . , S2.

In the algorithms described above, a technique may be applied to reduce the overall complexity. In particular, when a set of candidate constellations is generated and the performance of the candidate constellations are tested, those candidate constellations that have been previously tested (i.e. in one or more previous iteration) are not re-tested. That is, in a current iteration, only those candidate constellations that have not been tested in previous iterations are tested.

For example, as described above, a first set of candidate constellations, A, is generated in an iteration, and the best performing candidate constellation, a (a∈A), is selected from this set. In a next iteration, a second set of candidate constellations, B, is generated based on the previously selected constellation a (a∈B). In this next iteration, the best performing candidate constellation b (b∈B) from set B needs to be determined.

Typically, there will be at least some overlap between the two sets of candidate constellations A and B, such that one or more candidate constellations belong to both sets A and B (i.e. A∩B≠Ø), including constellation a. Since it is known that constellation a has the best performance of all the constellations in set A, then it is also known that constellation a has the best performance of all the constellations belonging to the overlap between sets A and B (i.e. A∩B).

Accordingly, when testing the constellations in set B to determine the best performing constellation, b, it is not necessary to re-test those constellations belonging to the overlap between sets A and B (i.e. it is not necessary to re-test those constellations in the set AnB). Instead, rather than testing all constellations in set B, only those constellations belonging to the smaller set of constellations B*, including constellations belonging to set B but excluding any constellations that also belong to set A (i.e. B*=B-A) are tested. Then, the best performing constellation from the set formed from the union of B* and the previous best performing constellation, a (i.e. the best performing constellation from the set B*∪{a}) is selected as the best performing constellation, b, of set B.

An example of the above principle in relation to the example shown in FIG. 16A is illustrated in FIG. 17 . In the example of FIG. 17 , at iteration i, it was found that the constellation point indicated as a black circle is the best performing constellation. At iteration i+1, there is no need to test the common subset (including the white circles and the black circle), because it was already tested before and gave an inferior performance. That is, at iteration i+1, only the dark grey circles need to be tested. Accordingly, in the illustrated example, a reduction in complexity of 44% (=4/9) is achieved.

FIG. 18 illustrates an apparatus for implementing an algorithm according to one or more of the embodiments described above. The apparatus is configured for generating a non-uniform constellation. The apparatus includes a block for performing a first process. The block for performing the first process includes: a block for obtaining a first constellation defined by one or more parameter values; and a block for generating a second constellation based on the first constellation using a second process. The block for generating the second constellation based on the first constellation using the second process includes: a block for obtaining a set of candidate constellations, wherein the set of candidate constellations includes the first constellation and one or more modified constellations, wherein each modified constellation is obtained by modifying the parameter values defining the first constellation; a block for determining performance of each candidate constellation according to a predetermined performance measure; and a block for selecting the candidate constellation having the best performance as the second constellation. The block for performing the first process further includes a block for determining a difference between the first constellation and the second constellation; and a block for, if the second constellation differs from the first constellation by more than a threshold amount, causing the block for performing the first process to repeat the first process using the second constellation generated in a current iteration of the first process as the first constellation in a next iteration.

The skilled person will appreciate that the functions of any two or more blocks illustrated in FIG. 18 may be performed by a single block, and that the functions of any block illustrated in FIG. 18 may be performed by two or more blocks. A block may be implemented in any suitable form, for example hardware, software, firmware, or any suitable combination of hardware, software and firmware.

A constellation obtained by a method according to the above exemplary embodiments may be used in a digital broadcasting system to transmit data from a transmitter side to a receiver side. In certain exemplary embodiments, the system includes a transmitter arranged to obtain data (e.g. a data stream), perform any required encoding and/or other processing of the data, modulate a signal using the data according to a modulation scheme corresponding to the constellation, and transmit the modulated signal. The system further includes a receiver configured to receive a modulated signal, demodulate the signal according to a demodulation scheme corresponding to the constellation (or a similar or corresponding constellation), and perform any necessary decoding and/or other processing to recover the original data. Certain exemplary embodiments may include a transmitter side apparatus only, a receiver side apparatus only, or a system including both a transmitter side apparatus and a receiver side apparatus.

In case of non-uniform constellations, it is possible to design constellations by relaxing only one constraint, i.e. by keeping a constellations square but changing a distance between constellation points. This form of non-uniform constellations (NUCs) can be referred to as one-dimensional (1D) NUCs. 1D NUCs can be described by the levels at which the constellations occur in the real positive part. The other points can be deduced by using the symmetry of the four quadrants as well as the real and imaginary symmetry. 1D-NUCs are simple to decode because of independence of the real and imaginary part. Two (2) pulse amplitude modulator (PAM) demappers can be used to decode 1D-NUCs.

It is possible to design a different type of NUC by relaxing both constraints: the square shape and the uniform distance between constellation points. Optimal constellations will have a tendency to look like a circular constellation. This type of NUC can be referred to as 2D-NUC. The 2D-NUC has a higher capacity than the 1D-NUC and a better BER/FER performance. However, performance of 2D-NUC comes at the expense of a more complex receiver demapper. Since the real and imaginary axes are not symmetrical, a 2D demapper is needed in order to decode a 2D-NUC constellation. In the case of 2D-NUC, a complete set of points need to be specified. It is possible to specify only the points belonging to the first quadrant and deduce the other points by supposing that the constellation is symmetrical.

Optimizing 1D and 2D NUCs depend on an SNR at which a capacity needs to be optimized. In the case of a BICM chain, an SNR can be selected to be a waterfall SNR of a BER/FER curve. The BER/FER waterfall can be defined as an SNR at which a BER curve falls below a certain level, for example 10e−6. The waterfall SNR depends on a coding rate of an LDPC encoder/decoder. As the code rate increases, the waterfall SNR increases. For this reason, a different NUC is associated with each LDPC encoder coding rate. The waterfall SNR also increases with a QAM constellation size (M). This is because the a receiver needs a higher SNR to decode a higher QAM constellation. Thus, a constellations size and a coding rate define the waterfall SNR. The waterfall SNR is used to optimize the constellations. Here, the coding rates include: 2/15, 3/15 and 4/15. The NUC sizes are: 16QAM, 64QAM, 256QAM and 1K QAM. For the first three QAM sizes only 2D constellations are proposed. For 1K QAM both 1D and 2D constellations are proposed.

Hereinafter, an example of constellation points of a constellation obtained by applying the algorithms described above by coding rates will be described.

In the following exemplary embodiment, a restriction is added to the process of determining capacity according to SNR with respect to the existing NUC designing method.

When SNR is given, it is general to calculate the maximum transmission capacity which can be transmitted with error-free. In other words, when calculating capacity by setting SNR with respect to BER/FER waterfall, the SNR indicates an area where a bit error or a frame error occurs, but actual capacity indicates transmission capacity under error-free circumstances, and thus, there may be a contradiction.

Therefore, in the present disclosure, when calculating capacity with respect to the SNR, a correction factor is added.

For example, when SNR1 is decided with respect to CH1 in FIG. 1 , if the capacity value under error-free state is C1, corrected C1 value, that is C1′ is defined as shown below. C1′=C1×(1−H(P _(b)))  [Equation 2]

Here, P_(b) indicates a BER value which determines a waterfall area, and function H(x) indicates binary entropy function, H(x)=−x×log₂(x)−(1−x)×log₂(1−x).

In this case, the reason why the value (1−H(P_(b))) which is equal to or smaller than 1 is multiplied to the existing capacity value as indicated in Equation 2 is as shown below.

With respect to a case where a bit error occurs as much as the probability P_(b) in a transmission channel, if it is assumed that a bit error as much as P_(b) has been occurred before transmitting source information, and error does not occur while source information is being transmitted, there is no difference in terms of a bit error in the final transmitting/receiving end. As described above, when the probability of error occurrence regarding a channel is considered a loss of source information, there is an effect as if lossy compression is applied as much as H(P_(b)) compared to given data, by the rate distortion theory of Shannon. That is, in conclusion, it can be considered that data amount which can be transmitted through firstly given channel, that is capacity, will be reduced compared to the channel (1−H(P_(b))) without an error or loss.

When the same value is applied to all the channels which consider P_(b) of FIG. 14 , the same factor(1−H(P_(b))) is multiplied, and thus, the same factor (1−H(P_(b))) is multiplied to the value of weighted capacity. Accordingly, there is no difference in the constellation points of the optimized NUC. However, target BER may be different for each channel, and therefore, the order of size of weighted capacity can be different. Accordingly, the constellation points of optimized NUC can be different. For example, an AWGN channel is applied to a fixed device in general, and thus, very low BER is required. Therefore, when calculating capacity, BER=1e−8 may be considered. As Rayleigh channel is largely considered for a mobile channel which experiences fading, BER higher than that of AWGN channel is required. Thus, BER=1e−6 can be considered. As described above, if different BER requirements are set for respective channels, values of factor for final capacity become different, and therefore, this value may be different from the NUC constellation points which are obtained in consideration of capacity only without reflecting BER.

To be specific, Table 2 indicates values of constellation points of a normalized 2D NU 16-QAM constellation (2D 16NUC) which is obtained by applying the algorithms described above using respective coding rates 2/15, 3/15 and 4/15 for a single SNR value.

TABLE 2 w/ Coding Rate Shape 2/15 3/15 4/15 w0  0.707316 + 0.707473i  0.709789 + 0.710216i  1.10696 + 0.580648i w1  0.707333 + 0.707367i  0.705044 + 0.710131i  0.581442 + 1.10703i w2  0.706049 + 0.707738i  0.709276 + 0.70389i  0.552546 + 0.362564i w3  0.706475 + 0.707102i  0.704489 + 0.703976i  0.362944 + 0.55265i w4 −0.707316 + 0.707473i −0.709789 + 0.710216i  −1.10696 + 0.580648i w5 −0.707333 + 0.707367i −0.705044 + 0.710131i −0.581442 + 1.10703i w6 −0.706049 + 0.707738i −0.709276 + 0.70389i −0.552546 + 0.362564i w7 −0.706475 + 0.707102i −0.704489 + 0.703976i −0.362944 + 0.55265i w8  0.707316 − 0.707473i  0.709789 − 0.710216i  1.10696 − 0.580648i w9  0.707333 − 0.707367i  0.705044 − 0.710131i  0.581442 − 1.10703i w10  0.706049 − 0.707738i  0.709276 − 0.70389i  0.552546 − 0.362564i w11  0.706475 − 0.707102i  0.704489 − 0.703976i  0.362944 − 0.55265i w12 −0.707316 − 0.707473i −0.709789 − 0.710216i  −1.10696 − 0.580648i w13 −0.707333 − 0.707367i −0.705044 − 0.710131i −0.581442 − 1.10703i w14 −0.706049 − 0.707738i −0.709276 − 0.70389i −0.552546 − 0.362564i w15 −0.706475 − 0.707102i −0.704489 − 0.703976i −0.362944 − 0.55265i

In this case, the constellation points of the 2D NU 16-QAM constellation for respective coding rates of 2/15, 3/15, and 4/15 are indicated in FIGS. 19-21 .

Table 3 indicates values of constellation points of a normalized 2D NU 64-QAM constellation (2D 64NUC) which is obtained by applying the algorithms described above using respective coding rates 2/15, 3/15 and 4/15 for a single SNR value.

TABLE 3 w/ Coding Rate Shape 2/15 3/15 4/15 w0   0.647424 + 0.983083i   0.547191 + 1.15913i    0.500831 +  1.21361i w1   0.643837 + 0.982885i   0.54734 + 1.15734i   0.499379 +  1.21941i w2   0.647063 + 0.97668i    0.546743 + 1.15987i    0.531316 +  1.17151i w3   0.644354 + 0.976223i   0.547937 + 1.15853i    0.529865 +  1.17877i w4   0.983862 + 0.647524i    1.15778 + 0.547787i    1.2107 + 0.503734i w5   0.977782 + 0.647422i    1.15763 + 0.547489i    1.22087 + 0.500831i w6   0.983498 + 0.643369i    1.15913 + 0.547489i    1.17151 + 0.529865i w7   0.977678 + 0.643265i    1.15913 + 0.547489i    1.18022 + 0.526961i w8   0.465916 + 0.639338i   0.316261 + 0.507211i   0.274368 + 0.476152i w9   0.464253 + 0.63861i    0.316261 + 0.507211i   0.272917 + 0.476152i w10   0.466124 + 0.635284i   0.316261 + 0.507211i   0.277272 + 0.479056i w11   0.463941 + 0.634972i   0.316261 + 0.507211i   0.277272 + 0.479056i w12   0.637769 + 0.46706i    0.508702 + 0.316261i   0.476152 + 0.272917i w13   0.635171 + 0.467267i   0.508702 + 0.316261i   0.476152 + 0.272917i w14   0.638497 + 0.465604i   0.508702 + 0.316261i   0.479056 + 0.277272i w15   0.635275 + 0.465293i   0.508702 + 0.316261i   0.479056 +  0.27582i w16 −0.647424 + 0.983083i −0.547191 + 1.15913i  −0.500831 +  1.21361i w17 −0.643837 + 0.982885i −0.54734 + 1.15734i −0.499379 +  1.21941i w18 −0.647063 + 0.97668i  −0.546743 + 1.15987i  −0.531316 +  1.17151i w19 −0.644354 + 0.976223i −0.547937 + 1.15853i  −0.529865 +  1.17877i w20 −0.983862 + 0.647524i  −1.15778 + 0.547787i  −1.2107 + 0.503734i w21 −0.977782 + 0.647422i  −1.15763 + 0.547489i  −1.22087 + 0.500831i w22 −0.983498 + 0.643369i  −1.15913 + 0.547489i  −1.17151 + 0.529865i w23 −0.977678 + 0.643265i  −1.15913 + 0.547489i  −1.18022 + 0.526961i w24 −0.465916 + 0.639338i −0.316261 + 0.507211i −0.274368 + 0.476152i w25 −0.464253 + 0.63861i  −0.316261 + 0.507211i −0.272917 + 0.476152i w26 −0.466124 + 0.635284i −0.316261 + 0.507211i −0.277272 + 0.479056i w27 −0.463941 + 0.634972i −0.316261 + 0.507211i −0.277272 + 0.479056i w28 −0.637769 + 0.46706i  −0.508702 + 0.316261i −0.476152 + 0.272917i w29 −0.635171 + 0.467267i −0.508702 + 0.316261i −0.476152 + 0.272917i w30 −0.638497 + 0.465604i −0.508702 + 0.316261i −0.479056 + 0.277272i w31 −0.635275 + 0.465293i −0.508702 + 0.316261i −0.479056 +  0.27582i w32   0.647424 − 0.983083i   0.547191 − 1.15913i    0.500831 −  1.21361i w33   0.643837 − 0.982885i  0.54734 − 1.15734i   0.499379 −  1.21941i w34   0.647063 − 0.97668i    0.546743 − 1.15987i    0.531316 −  1.17151i w35   0.644354 − 0.976223i   0.547937 − 1.15853i    0.529865 −  1.17877i w36   0.983862 − 0.647524i    1.15778 − 0.547787i    1.2107 − 0.503734i w37   0.977782 − 0.647422i    1.15763 − 0.547489i    1.22087 − 0.500831i w38   0.983498 − 0.643369i    1.15913 − 0.547489i    1.17151 − 0.529865i w39   0.977678 − 0.643265i    1.15913 − 0.547489i    1.18022 − 0.526961i w40   0.465916 − 0.639338i   0.316261 − 0.507211i   0.274368 − 0.476152i w41   0.464253 − 0.63861i    0.316261 − 0.507211i   0.272917 − 0.476152i w42   0.466124 − 0.635284i   0.316261 − 0.507211i   0.277272 − 0.479056i w43   0.463941 − 0.634972i   0.316261 − 0.507211i   0.277272 − 0.479056i w44   0.637769 − 0.46706i    0.508702 − 0.316261i   0.476152 − 0.272917i w45   0.635171 − 0.467267i   0.508702 − 0.316261i   0.476152 − 0.272917i w46   0.638497 − 0.465604i   0.508702 − 0.316261i   0.479056 − 0.277272i w47   0.635275 − 0.465293i   0.508702 − 0.316261i   0.479056 −  0.27582i  w48 −0.647424 − 0.983083i −0.547191 − 1.15913i  −0.500831 −  1.21361i  w49 −0.643837 − 0.982885i −0.54734 − 1.15734i −0.499379 −  1.21941i  w50 −0.647063 − 0.97668i  −0.546743 − 1.15987i  −0.531316 −  1.17151i  w51 −0.644354 − 0.976223i −0.547937 − 1.15853i  −0.529865 −  1.17877i  w52 −0.983862 − 0.647524i  −1.15778 − 0.547787i  −1.2107 − 0.503734i w53 −0.977782 − 0.647422i  −1.15763 − 0.547489i  −1.22087 − 0.500831i w54 −0.983498 − 0.643369i  −1.15913 − 0.547489i  −1.17151 − 0.529865i w55 −0.977678 − 0.643265i  −1.15913 − 0.547489i  −1.18022 − 0.526961i w56 −0.465916 − 0.639338i −0.316261 − 0.507211i −0.274368 − 0.476152i w57 −0.464253 − 0.63861i  −0.316261 − 0.507211i −0.272917 − 0.476152i w58 −0.466124 − 0.635284i −0.316261 − 0.507211i −0.277272 − 0.479056i w59 −0.463941 − 0.634972i −0.316261 − 0.507211i −0.277272 − 0.479056i w60 −0.637769 − 0.46706i  −0.508702 − 0.316261i −0.476152 − 0.272917i w61 −0.635171 − 0.467267i −0.508702 − 0.316261i −0.476152 − 0.272917i w62 −0.638497 − 0.465604i −0.508702 − 0.316261i −0.479056 − 0.277272i w63 −0.635275 − 0.465293i −0.508702 − 0.316261i −0.479056 − 0.27582i 

In this case, the constellation points of the 2D NU 64-QAM constellation for respective coding rates of 2/15, 3/15 and 4/15 are indicated in FIGS. 22-24 .

Table 4 indicates values of constellation points of a normalized 2D NU 256-QAM constellation (2D 256NUC) which is obtained by applying the algorithms described above using respective coding rates 2/15, 3/15 and 4/15 for a single SNR value.

TABLE 4 w/ Coding Rate Shape 2/15 3/15 4/15 w0  0.555322 + 1.12624i   0.522922 + 1.18101i    0.297463 +  1.05643i w1   0.56728 + 1.13359i   0.538432 + 1.16254i    0.586177 +  0.96165i w2  0.559308 + 1.12037i   0.514797 + 1.1943i    0.290901 +  1.06955i w3  0.563576 + 1.13205i   0.528831 + 1.1751i    0.579615 + 0.968941i w4  0.552505 + 1.12494i   0.498548 + 1.22015i    0.295276 +  1.33567i w5  0.563704 + 1.13202i   0.511105 + 1.19725i     0.74876 +  1.23651i w6  0.559846 + 1.11805i   0.488947 + 1.23566i    0.300379 +  1.51137i w7  0.565901 + 1.1274i    0.504457 + 1.21129i    0.815106 +  1.3816i w8  0.557904 + 1.1381i    0.522183 + 1.18174i    0.300379 +  1.05351i w9  0.561681 + 1.14706i   0.536955 + 1.16402i    0.584718 + 0.963109i w10 0.559343 + 1.13456i   0.51332 + 1.19504i   0.293088 +  1.06591i w11 0.567186 + 1.14299i   0.530308 + 1.1751i    0.582531 + 0.966754i w12 0.553299 + 1.13551i   0.497071 + 1.22163i    0.295276 +  1.3189i w13 0.563201 + 1.14214i   0.512582 + 1.19947i    0.746573 +  1.21683i w14 0.556748 + 1.13251i   0.488208 + 1.23714i    0.296005 +  1.46544i w15 0.564101 + 1.13628i   0.504457 + 1.21277i    0.829688 +  1.35389i w16    1.13089 + 0.559658i    1.17953 + 0.525138i    1.06372 + 0.296005i w17    1.14052 + 0.565983i    1.16254 + 0.538432i    0.96165 + 0.581073i w18    1.13482 + 0.558817i   1.19135 + 0.51332i    1.0732 + 0.293088i w19    1.14905 + 0.563844i    1.17436 + 0.529569i   0.968212 + 0.581802i w20    1.12445 + 0.561493i    1.22089 + 0.499287i    1.36191 +  0.29965i w21    1.13332 + 0.562675i    1.20021 + 0.514797i    1.22485 + 0.754593i w22    1.12837 + 0.557788i    1.23418 + 0.488208i    1.54272 + 0.310586i w23    1.14364 + 0.563576i    1.21424 + 0.505196i    1.39691 + 0.852289i w24    1.11959 + 0.562018i    1.18027 + 0.522922i    1.06153 + 0.294546i w25    1.13468 + 0.566507i   1.16402 + 0.53991i   0.963109 + 0.581802i w26   1.13792 + 0.56113i   1.19208 + 0.51332i    1.07101 + 0.292359i w27    1.14396 + 0.563832i    1.17584 + 0.530308i   0.967483 + 0.582531i w28    1.12214 + 0.559424i    1.22089 + 0.497071i    1.32546 + 0.297463i w29    1.13182 + 0.568564i    1.20243 + 0.514797i    1.19787 + 0.749489i w30   1.13018 + 0.56189i    1.23492 + 0.488947i    1.45596 + 0.304024i w31    1.13855 + 0.566158i    1.21498 + 0.504457i    1.32692 + 0.841353i w32   0.33943 + 0.53805i   0.274017 + 0.477129i   0.249344 + 0.558472i w33   0.339746 + 0.53604i    0.276233 + 0.480084i   0.296005 + 0.534412i w34    0.33867 + 0.532417i   0.273278 + 0.475652i   0.244969 + 0.541703i w35   0.340014 + 0.533493i   0.274756 + 0.477868i   0.287256 + 0.519102i w36   0.337373 + 0.530582i   0.270324 + 0.474175i    0.20487 + 0.392243i w37   0.340506 + 0.534252i    0.27254 + 0.476391i   0.217264 + 0.380577i w38   0.337911 + 0.532417i   0.269585 + 0.472698i   0.199038 + 0.375474i w39   0.340014 + 0.531704i   0.271801 + 0.474914i   0.210703 + 0.364538i w40   0.339652 + 0.536975i   0.274017 + 0.477868i   0.249344 +  0.55993i w41   0.339968 + 0.53832i    0.275494 + 0.479345i   0.297463 + 0.535141i w42   0.338133 + 0.534696i    0.27254 + 0.475652i   0.244969 +  0.54389i w43   0.338179 + 0.534743i   0.274756 + 0.477868i   0.288714 + 0.521289i w44   0.337864 + 0.534158i   0.271063 + 0.473436i   0.205599 + 0.393701i w45   0.338939 + 0.533177i    0.27254 + 0.476391i   0.218723 + 0.382035i w46    0.34019 + 0.534696i   0.269585 + 0.471959i   0.199767 + 0.376203i w47   0.338449 + 0.533983i   0.271063 + 0.474175i   0.212161 + 0.366725i w48   0.534965 + 0.33943i    0.477129 + 0.274017i   0.560659 + 0.248615i w49   0.536309 + 0.339746i   0.478607 + 0.276233i   0.538058 + 0.296005i w50   0.534205 + 0.338939i   0.476391 + 0.27254i     0.54389 +  0.24424i w51   0.538366 + 0.337957i   0.477129 + 0.274756i   0.522018 + 0.286527i w52   0.532907 + 0.336344i   0.473436 + 0.270324i   0.390784 +  0.20487i w53   0.532955 + 0.338717i   0.475652 + 0.27254i    0.381306 + 0.217264i w54   0.53112 + 0.338939i   0.473436 + 0.269585i   0.374016 + 0.199767i w55   0.533223 + 0.337957i   0.474175 + 0.271063i   0.365267 + 0.209974i w56   0.531342 + 0.339652i   0.477129 + 0.274017i   0.564304 + 0.248615i w57   0.532417 + 0.339968i   0.477868 + 0.276233i   0.540974 + 0.296734i w58   0.533936 + 0.34019i    0.476391 + 0.27254i    0.547536 + 0.243511i w59    0.53604 + 0.340506i   0.477129 + 0.274756i   0.525663 + 0.287985i w60   0.528525 + 0.339652i   0.474175 + 0.270324i   0.393701 +  0.20487i w61   0.531658 + 0.337911i   0.474914 + 0.27254i    0.384952 + 0.218723i w62   0.531879 + 0.338133i   0.473436 + 0.269585i   0.376203 + 0.199767i w63   0.532686 + 0.339477i   0.474914 + 0.271063i   0.368912 + 0.211432i w64 −0.555322 + 1.12624i  −0.522922 + 1.18101i  −0.297463 +  1.05643i w65 −0.56728 + 1.13359i −0.538432 + 1.16254i  −0.586177 +  0.96165i w66 −0.559308 + 1.12037i  −0.514797 + 1.1943i  −0.290901 +  1.06955i w67 −0.563576 + 1.13205i  −0.528831 + 1.1751i  −0.579615 + 0.968941i w68 −0.552505 + 1.12494i  −0.498548 + 1.22015i  −0.295276 +  1.33567i w69 −0.563704 + 1.13202i  −0.511105 + 1.19725i   −0.74876 +  1.23651i w70 −0.559846 + 1.11805i  −0.488947 + 1.23566i  −0.300379 +  1.51137i w71 −0.565901 + 1.1274i  −0.504457 + 1.21129i  −0.815106 +  1.3816i w72 −0.557904 + 1.1381i  −0.522183 + 1.18174i  −0.300379 +  1.05351i w73 −0.561681 + 1.14706i  −0.536955 + 1.16402i  −0.584718 + 0.963109i w74 −0.559343 + 1.13456i  −0.51332 + 1.19504i −0.293088 +  1.06591i w75 −0.567186 + 1.14299i  −0.530308 + 1.1751i  −0.582531 + 0.966754i w76 −0.553299 + 1.13551i  −0.497071 + 1.22163i  −0.295276 +  1.3189i w77 −0.563201 + 1.14214i  −0.512582 + 1.19947i  −0.746573 +  1.21683i w78 −0.556748 + 1.13251i  −0.488208 + 1.23714i  −0.296005 +  1.46544i w79 −0.564101 + 1.13628i  −0.504457 + 1.21277i  −0.829688 +  1.35389i w80  −1.13089 + 0.559658i  −1.17953 + 0.525138i  −1.06372 + 0.296005i w81  −1.14052 + 0.565983i  −1.16254 + 0.538432i  −0.96165 + 0.581073i w82  −1.13482 + 0.558817i −1.19135 + 0.51332i  −1.0732 + 0.293088i w83  −1.14905 + 0.563844i  −1.17436 + 0.529569i −0.968212 + 0.581802i w84  −1.12445 + 0.561493i  −1.22089 + 0.499287i  −1.36191 +  0.29965i w85  −1.13332 + 0.562675i  −1.20021 + 0.514797i  −1.22485 + 0.754593i w86  −1.12837 + 0.557788i  −1.23418 + 0.488208i  −1.54272 + 0.310586i w87  −1.14364 + 0.563576i  −1.21424 + 0.505196i  −1.39691 + 0.852289i w88  −1.11959 + 0.562018i  −1.18027 + 0.522922i  −1.06153 + 0.294546i w89  −1.13468 + 0.566507i −1.16402 + 0.53991i −0.963109 + 0.581802i w90 −1.13792 + 0.56113i −1.19208 + 0.51332i  −1.07101 + 0.292359i w91  −1.14396 + 0.563832i  −1.17584 + 0.530308i −0.967483 + 0.582531i w92  −1.12214 + 0.559424i  −1.22089 + 0.497071i  −1.32546 + 0.297463i w93  −1.13182 + 0.568564i  −1.20243 + 0.514797i  −1.19787 + 0.749489i w94 −1.13018 + 0.56189i  −1.23492 + 0.488947i  −1.45596 + 0.304024i w95  −1.13855 + 0.566158i  −1.21498 + 0.504457i  −1.32692 + 0.841353i w96 −0.33943 + 0.53805i −0.274017 + 0.477129i −0.249344 + 0.558472i w97 −0.339746 + 0.53604i  −0.276233 + 0.480084i −0.296005 + 0.534412i w98  −0.33867 + 0.532417i −0.273278 + 0.475652i −0.244969 + 0.541703i w99 −0.340014 + 0.533493i −0.274756 + 0.477868i −0.287256 + 0.519102i  w100 −0.337373 + 0.530582i −0.270324 + 0.474175i  −0.20487 + 0.392243i  w101 −0.340506 + 0.534252i  −0.27254 + 0.476391i −0.217264 + 0.380577i  w102 −0.337911 + 0.532417i −0.269585 + 0.472698i −0.199038 + 0.375474i  w103 −0.340014 + 0.531704i −0.271801 + 0.474914i −0.210703 + 0.364538i  w104 −0.339652 + 0.536975i −0.274017 + 0.477868i −0.249344 +  0.55993i  w105 −0.339968 + 0.53832i  −0.275494 + 0.479345i −0.297463 + 0.535141i  w106 −0.338133 + 0.534696i  −0.27254 + 0.475652i −0.244969 +  0.54389i  w107 −0.338179 + 0.534743i −0.274756 + 0.477868i −0.288714 + 0.521289i  w108 −0.337864 + 0.534158i −0.271063 + 0.473436i −0.205599 + 0.393701i  w109 −0.338939 + 0.533177i  −0.27254 + 0.476391i −0.218723 + 0.382035i  w110  −0.34019 + 0.534696i −0.269585 + 0.471959i −0.199767 + 0.376203i  w111 −0.338449 + 0.533983i −0.271063 + 0.474175i −0.212161 + 0.366725i  w112 −0.534965 + 0.33943i  −0.477129 + 0.274017i −0.560659 + 0.248615i  w113 −0.536309 + 0.339746i −0.478607 + 0.276233i −0.538058 + 0.296005i  w114 −0.534205 + 0.338939i −0.476391 + 0.27254i   −0.54389 +  0.24424i  w115 −0.538366 + 0.337957i −0.477129 + 0.274756i −0.522018 + 0.286527i  w116 −0.532907 + 0.336344i −0.473436 + 0.270324i −0.390784 +  0.20487i  w117 −0.532955 + 0.338717i −0.475652 + 0.27254i  −0.381306 + 0.217264i  w118  −0.53112 + 0.338939i −0.473436 + 0.269585i −0.374016 + 0.199767i  w119 −0.533223 + 0.337957i −0.474175 + 0.271063i −0.365267 + 0.209974i  w120 −0.531342 + 0.339652i −0.477129 + 0.274017i −0.564304 + 0.248615i  w121 −0.532417 + 0.339968i −0.477868 + 0.276233i −0.540974 + 0.296734i  w122 −0.533936 + 0.34019i  −0.476391 + 0.27254i  −0.547536 + 0.243511i  w123  −0.53604 + 0.340506i −0.477129 + 0.274756i −0.525663 + 0.287985i  w124 −0.528525 + 0.339652i −0.474175 + 0.270324i −0.393701 +  0.20487i  w125 −0.531658 + 0.337911i −0.474914 + 0.27254i  −0.384952 + 0.218723i  w126 −0.531879 + 0.338133i −0.473436 + 0.269585i −0.376203 + 0.199767i  w127 −0.532686 + 0.339477i −0.474914 + 0.271063i −0.368912 + 0.211432i  w128   0.555322 − 1.12624i    0.522922 − 1.18101i    0.297463 −  1.05643i  w129    0.56728 − 1.13359i    0.538432 − 1.16254i    0.586177 −  0.96165i  w130   0.559308 − 1.12037i    0.514797 − 1.1943i    0.290901 −  1.06955i  w131   0.563576 − 1.13205i    0.528831 − 1.1751i    0.579615 − 0.968941i  w132   0.552505 − 1.12494i    0.498548 − 1.22015i    0.295276 −  1.33567i  w133   0.563704 − 1.13202i    0.511105 − 1.19725i     0.74876 −  1.23651i  w134   0.559846 − 1.11805i    0.488947 − 1.23566i    0.300379 −  1.51137i  w135   0.565901 − 1.1274i    0.504457 − 1.21129i    0.815106 −  1.3816i  w136   0.557904 − 1.1381i    0.522183 − 1.18174i    0.300379 −  1.05351i  w137   0.561681 − 1.14706i    0.536955 − 1.16402i    0.584718 − 0.963109i  w138   0.559343 − 1.13456i    0.51332 − 1.19504i   0.293088 −  1.06591i  w139   0.567186 − 1.14299i    0.530308 − 1.1751i    0.582531 − 0.966754i  w140   0.553299 − 1.13551i    0.497071 − 1.22163i    0.295276 −  1.3189i  w141   0.563201 − 1.14214i    0.512582 − 1.19947i    0.746573 −  1.21683i  w142   0.556748 − 1.13251i    0.488208 − 1.23714i    0.296005 −  1.46544i  w143   0.564101 − 1.13628i    0.504457 − 1.21277i    0.829688 −  1.35389i  w144    1.13089 − 0.559658i    1.17953 − 0.525138i    1.06372 − 0.296005i  w145    1.14052 − 0.565983i    1.16254 − 0.538432i    0.96165 − 0.581073i  w146    1.13482 − 0.558817i   1.19135 − 0.51332i    1.0732 − 0.293088i  w147    1.14905 − 0.563844i    1.17436 − 0.529569i   0.968212 − 0.581802i  w148    1.12445 − 0.561493i    1.22089 − 0.499287i    1.36191 −  0.29965i  w149    1.13332 − 0.562675i    1.20021 − 0.514797i    1.22485 − 0.754593i  w150    1.12837 − 0.557788i    1.23418 − 0.488208i    1.54272 − 0.310586i  w151    1.14364 − 0.563576i    1.21424 − 0.505196i    1.39691 − 0.852289i  w152    1.11959 − 0.562018i    1.18027 − 0.522922i    1.06153 − 0.294546i  w153    1.13468 − 0.566507i   1.16402 − 0.53991i   0.963109 − 0.581802i  w154   1.13792 − 0.56113i   1.19208 − 0.51332i    1.07101 − 0.292359i  w155    1.14396 − 0.563832i    1.17584 − 0.530308i   0.967483 − 0.582531i  w156    1.12214 − 0.559424i    1.22089 − 0.497071i    1.32546 − 0.297463i  w157    1.13182 − 0.568564i    1.20243 − 0.514797i    1.19787 − 0.749489i  w158   1.13018 − 0.56189i    1.23492 − 0.488947i    1.45596 − 0.304024i  w159    1.13855 − 0.566158i    1.21498 − 0.504457i    1.32692 − 0.841353i  w160   0.33943 − 0.53805i   0.274017 − 0.477129i   0.249344 − 0.558472i  w161   0.339746 − 0.53604i    0.276233 − 0.480084i   0.296005 − 0.534412i  w162    0.33867 − 0.532417i   0.273278 − 0.475652i   0.244969 − 0.541703i  w163   0.340014 − 0.533493i   0.274756 − 0.477868i   0.287256 − 0.519102i  w164   0.337373 − 0.530582i   0.270324 − 0.474175i    0.20487 − 0.392243i  w165   0.340506 − 0.534252i    0.27254 − 0.476391i   0.217264 − 0.380577i  w166   0.337911 − 0.532417i   0.269585 − 0.472698i   0.199038 − 0.375474i  w167   0.340014 − 0.531704i   0.271801 − 0.474914i   0.210703 − 0.364538i  w168   0.339652 − 0.536975i   0.274017 − 0.477868i   0.249344 −  0.55993i  w169   0.339968 − 0.53832i    0.275494 − 0.479345i   0.297463 − 0.535141i  w170   0.338133 − 0.534696i    0.27254 − 0.475652i   0.244969 −  0.54389i  w171   0.338179 − 0.534743i   0.274756 − 0.477868i   0.288714 − 0.521289i  w172   0.337864 − 0.534158i   0.271063 − 0.473436i   0.205599 − 0.393701i  w173   0.338939 − 0.533177i    0.27254 − 0.476391i   0.218723 − 0.382035i  w174    0.34019 − 0.534696i   0.269585 − 0.471959i   0.199767 − 0.376203i  w175   0.338449 − 0.533983i   0.271063 − 0.474175i   0.212161 − 0.366725i  w176   0.534965 − 0.33943i    0.477129 − 0.274017i   0.560659 − 0.248615i  w177   0.536309 − 0.339746i   0.478607 − 0.276233i   0.538058 − 0.296005i  w178   0.534205 − 0.338939i   0.476391 − 0.27254i     0.54389 −  0.24424i  w179   0.538366 − 0.337957i   0.477129 − 0.274756i   0.522018 − 0.286527i  w180   0.532907 − 0.336344i   0.473436 − 0.270324i   0.390784 −  0.20487i  w181   0.532955 − 0.338717i   0.475652 − 0.27254i    0.381306 − 0.217264i  w182    0.53112 − 0.338939i   0.473436 − 0.269585i   0.374016 − 0.199767i  w183   0.533223 − 0.337957i   0.474175 − 0.271063i   0.365267 − 0.209974i  w184   0.531342 − 0.339652i   0.477129 − 0.274017i   0.564304 − 0.248615i  w185   0.532417 − 0.339968i   0.477868 − 0.276233i   0.540974 − 0.296734i  w186   0.533936 − 0.34019i    0.476391 − 0.27254i    0.547536 − 0.243511i  w187    0.53604 − 0.340506i   0.477129 − 0.274756i   0.525663 − 0.287985i  w188   0.528525 − 0.339652i   0.474175 − 0.270324i   0.393701 −  0.20487i  w189   0.531658 − 0.337911i   0.474914 − 0.27254i    0.384952 − 0.218723i  w190   0.531879 − 0.338133i   0.473436 − 0.269585i   0.376203 − 0.199767i  w191   0.532686 − 0.339477i   0.474914 − 0.271063i   0.368912 − 0.211432i  w192 −0.555322 − 1.12624i  −0.522922 − 1.18101i  −0.297463 −  1.05643i  w193 −0.56728 − 1.13359i −0.538432 − 1.16254i  −0.586177 −  0.96165i  w194 −0.559308 − 1.12037i  −0.514797 − 1.1943i  −0.290901 −  1.06955i  w195 −0.563576 − 1.13205i  −0.528831 − 1.1751i  −0.579615 − 0.968941i  w196 −0.552505 − 1.12494i  −0.498548 − 1.22015i  −0.295276 −  1.33567i  w197 −0.563704 − 1.13202i  −0.511105 − 1.19725i   −0.74876 −  1.23651i  w198 −0.559846 − 1.11805i  −0.488947 − 1.23566i  −0.300379 −  1.51137i  w199 −0.565901 − 1.1274i  −0.504457 − 1.21129i  −0.815106 −  1.3816i  w200 −0.557904 − 1.1381i  −0.522183 − 1.18174i  −0.300379 −  1.05351i  w201 −0.561681 − 1.14706i  −0.536955 − 1.16402i  −0.584718 − 0.963109i  w202 −0.559343 − 1.13456i  −0.51332 − 1.19504i −0.293088 −  1.06591i  w203 −0.567186 − 1.14299i  −0.530308 − 1.1751i  −0.582531 − 0.966754i  w204 −0.553299 − 1.13551i  −0.497071 − 1.22163i  −0.295276 −  1.3189i  w205 −0.563201 − 1.14214i  −0.512582 − 1.19947i  −0.746573 −  1.21683i  w206 −0.556748 − 1.13251i  −0.488208 − 1.23714i  −0.296005 −  1.46544i  w207 −0.564101 − 1.13628i  −0.504457 − 1.21277i  −0.829688 −  1.35389i  w208  −1.13089 − 0.559658i  −1.17953 − 0.525138i  −1.06372 − 0.296005i  w209  −1.14052 − 0.565983i  −1.16254 − 0.538432i  −0.96165 − 0.581073i  w210  −1.13482 − 0.558817i −1.19135 − 0.51332i  −1.0732 − 0.293088i  w211  −1.14905 − 0.563844i  −1.17436 − 0.529569i −0.968212 − 0.581802i  w212  −1.12445 − 0.561493i  −1.22089 − 0.499287i  −1.36191 −  0.29965i  w213  −1.13332 − 0.562675i  −1.20021 − 0.514797i  −1.22485 − 0.754593i  w214  −1.12837 − 0.557788i  −1.23418 − 0.488208i  −1.54272 − 0.310586i  w215  −1.14364 − 0.563576i  −1.21424 − 0.505196i  −1.39691 − 0.852289i  w216  −1.11959 − 0.562018i  −1.18027 − 0.522922i  −1.06153 − 0.294546i  w217  −1.13468 − 0.566507i −1.16402 − 0.53991i −0.963109 − 0.581802i  w218 −1.13792 − 0.56113i −1.19208 − 0.51332i  −1.07101 − 0.292359i  w219  −1.14396 − 0.563832i  −1.17584 − 0.530308i −0.967483 − 0.582531i  w220  −1.12214 − 0.559424i  −1.22089 − 0.497071i  −1.32546 − 0.297463i  w221  −1.13182 − 0.568564i  −1.20243 − 0.514797i  −1.19787 − 0.749489i  w222 −1.13018 − 0.56189i  −1.23492 − 0.488947i  −1.45596 − 0.304024i  w223  −1.13855 − 0.566158i  −1.21498 − 0.504457i  −1.32692 − 0.841353i  w224 −0.33943 − 0.53805i −0.274017 − 0.477129i −0.249344 − 0.558472i  w225 −0.339746 − 0.53604i  −0.276233 − 0.480084i −0.296005 − 0.534412i  w226  −0.33867 − 0.532417i −0.273278 − 0.475652i −0.244969 − 0.541703i  w227 −0.340014 − 0.533493i −0.274756 − 0.477868i −0.287256 − 0.519102i  w228 −0.337373 − 0.530582i −0.270324 − 0.474175i  −0.20487 − 0.392243i  w229 −0.340506 − 0.534252i  −0.27254 − 0.476391i −0.217264 − 0.380577i  w230 −0.337911 − 0.532417i −0.269585 − 0.472698i −0.199038 − 0.375474i  w231 −0.340014 − 0.531704i −0.271801 − 0.474914i −0.210703 − 0.364538i  w232 −0.339652 − 0.536975i −0.274017 − 0.477868i −0.249344 −  0.55993i  w233 −0.339968 − 0.53832i  −0.275494 − 0.479345i −0.297463 − 0.535141i  w234 −0.338133 − 0.534696i  −0.27254 − 0.475652i −0.244969 −  0.54389i  w235 −0.338179 − 0.534743i −0.274756 − 0.477868i −0.288714 − 0.521289i  w236 −0.337864 − 0.534158i −0.271063 − 0.473436i −0.205599 − 0.393701i  w237 −0.338939 − 0.533177i  −0.27254 − 0.476391i −0.218723 − 0.382035i  w238  −0.34019 − 0.534696i −0.269585 − 0.471959i −0.199767 − 0.376203i  w239 −0.338449 − 0.533983i −0.271063 − 0.474175i −0.212161 − 0.366725i  w240 −0.534965 − 0.33943i  −0.477129 − 0.274017i −0.560659 − 0.248615i  w241 −0.536309 − 0.339746i −0.478607 − 0.276233i −0.538058 − 0.296005i  w242 −0.534205 − 0.338939i −0.476391 − 0.27254i   −0.54389 −  0.24424i  w243 −0.538366 − 0.337957i −0.477129 − 0.274756i −0.522018 − 0.286527i  w244 −0.532907 − 0.336344i −0.473436 − 0.270324i −0.390784 −  0.20487i  w245 −0.532955 − 0.338717i −0.475652 − 0.27254i  −0.381306 − 0.217264i  w246  −0.53112 − 0.338939i −0.473436 − 0.269585i −0.374016 − 0.199767i  w247 −0.533223 − 0.337957i −0.474175 − 0.271063i −0.365267 − 0.209974i  w248 −0.531342 − 0.339652i −0.477129 − 0.274017i −0.564304 − 0.248615i  w249 −0.532417 − 0.339968i −0.477868 − 0.276233i −0.540974 − 0.296734i  w250 −0.533936 − 0.34019i  −0.476391 − 0.27254i  −0.547536 − 0.243511i  w251  −0.53604 − 0.340506i −0.477129 − 0.274756i −0.525663 − 0.287985i  w252 −0.528525 − 0.339652i −0.474175 − 0.270324i −0.393701 −  0.20487i  w253 −0.531658 − 0.337911i −0.474914 − 0.27254i  −0.384952 − 0.218723i  w254 −0.531879 − 0.338133i −0.473436 − 0.269585i −0.376203 − 0.199767i  w255 −0.532686 − 0.339477i −0.474914 − 0.271063i −0.368912 − 0.211432i

In this case, the constellation points of the 2D NU 256-QAM constellation for the respective coding rates of 2/15, 3/15, and 4/15 are illustrated in FIGS. 25-27 .

Meanwhile, according to Tables 2-4, when values of constellation points are determined in one quadrant, values of constellation points in other quadrants may be deduced by symmetry. For example, for each constellation point A in the top-right quadrant, corresponding constellation points may be present in three different quadrants (bottom-right, bottom-left and top-left) respectively, and they can be indicated as A*, −A*, and −A. Here, * indicates complex conjugation.

Table 5 indicates values of constellation points of a normalized 1D NU 1024-QAM constellation (1D 1024NUC) which is obtained by applying the algorithms described above using respective coding rates 2/15, 3/15 and 4/15 for a single SNR value.

TABLE 5 Coding Rate Level 2/15 3/15 4/15 1 1 1 1 2 1.000988842 1.073113208 1.008229665 3 1.001438042 1.153930818 1.075789474 4 1.001284289 1.073899371 1.065263158 5 1.002909247 1.153459119 1.588516746 6 1.00312028 1.238050314 1.608803828 7 1.00152547 1.153930818 1.502200957 8 1.0015315 1.073742138 1.483062201 9 2.824341802 3.982232704 3.724784689 10 2.839150319 3.504402516 3.826411483 11 2.868001604 3.278616352 3.897799043 12 2.855026063 3.504559748 3.807272727 13 2.848080048 3.972484277 5.023923445 14 2.861384198 3.499371069 5.023349282 15 2.831923931 3.970754717 5.88937799 16 2.818221832 6.002830189 7.67138756

In this case, the constellation points of the 1D NU 1K QAM constellation for the respective coding rates 2/15, 3/15 and 4/15 may be illustrated in FIGS. 28-30 .

For example, FIG. 28 illustrates an exemplary 1D NU 1024-QAM constellation obtained by applying the algorithms described above using the coding rate of 2/15.

According to FIG. 28 , a complete set of constellation points are indicated in a constellation diagram on the right-hand side of the drawing. Values of the constellation points of the top-right quadrant are indicated on the top-left side of the drawing.

In the case of the 1D NU 1K QAM constellation, rather than giving the values of the constellation points explicitly, a set of levels of the constellation points are given instead, from which actual values of the constellation points may be deduced. To be specific, given a set of m levels A=[A₁, A₂, . . . , A_(m)], a set of m² constellation point values C+Di may be deduced. Herein, C and D each may include a value selected from a level set A. A complete set of constellation points in the top-right quadrant may be obtained by considering all possible pairs of values C and D. According to FIG. 28 , values of constellation points in the remaining three quadrants may be similarly deduced by symmetry. As an example, according to Table 5, when the coding rate is 2/15, A={1, 1.000988842, . . . , 2.818221832}, and a group of C+Di corresponding to a constellation point set of the first quadrant has 256 elements such as {1+i, 1+1.000988842×i, 1.000988842+i, . . . , 2.818221832+2.818221832×i}, the complete set of 1D NU 1024-QAM constellation points may be obtained by indicating an arbitrary element a in the group, as a*, −a* and −a. Here, * indicates complex conjugation.

Table 6 indicates values of constellation points of a normalized 1D NU 4096-QAM constellation (1D 4096NUC) which is obtained by applying the algorithms described above using respective coding rates 2/15, 3/15, 4/15 and 5/15 for a single SNR value.

TABLE 6 Coding Rate Level 2/15 3/15 4/15 5/15 1 1 1 1 1 2 1.020833333 1 0.973180077 1 3 1.041666667 1.057142857 0.965517241 1 4 1.020833333 1.057142857 0.980842912 1 5 1.041666667 1.057142857 0.988505747 1 6 1.0625 1.057142857 0.957854406 1 7 1.041666667 1.028571429 0.988505747 1 8 1.020833333 1 1 1 9 1.041666667 1.371428571 2.233716475 2.863636364 10 1.0625 1.428571429 2.233716475 2.863636364 11 1.0625 1.485714286 2.210727969 2.772727273 12 1.0625 1.457142857 2.229885057 2.772727273 13 1.041666667 1.457142857 2.245210728 2.772727273 14 1.0625 1.485714286 2.22605364 2.772727273 15 1.041666667 1.371428571 2.214559387 2.863636364 16 1.020833333 1.371428571 2.237547893 2.863636364 17 3.4375 3.542857143 4.275862069 4.863636364 18 3.041666667 3.628571429 4.356321839 4.863636364 19 2.791666667 3.657142857 4.708812261 5.545454545 20 3 3.6 4.62835249 5.590909091 21 2.916666667 3.628571429 4.624521073 5.545454545 22 2.75 3.685714286 4.724137931 5.545454545 23 2.958333333 3.657142857 4.360153257 4.863636364 24 3.25 3.6 4.287356322 4.863636364 25 4.125 6.4 8.98467433 11.18181818 26 3.354166667 5.314285714 7.911877395 9.772727273 27 2.979166667 4.8 6.708812261 8 28 3.291666667 4.971428571 6.536398467 7.545454545 29 3.520833333 4.857142857 6.425287356 7.5 30 3.083333333 4.714285714 6.609195402 7.909090909 31 3.604166667 5.228571429 7.747126437 9.454545455 32 5.1875 8.057142857 10.8697318 13.18181818

In this case, the constellation points of the 1D NU 4096-QAM constellation for the respective coding rates 2/15, 3/15, 4/15 and 5/15 may be illustrated in FIGS. 31-34 .

In the case of the 1D NU 4K QAM constellation, rather than giving the values of the constellation points explicitly, a set of levels of the constellation points are given instead, from which actual values of the constellation points may be deduced. To be specific, given a set of m levels A=[A₁, A₂, . . . , A_(m)], and a set of m² constellation point values C+Di may be deduced. Herein, C and D each may include a value selected from a level set A. A complete set of constellation points in the top-right quadrant may be obtained by considering all possible pairs of values C and D. That is, values of constellation points in the remaining three quadrants may be similarly deduced by symmetry. As an example, according to Table 6, when the coding rate is 2/15, A={1, 1.020833333, . . . , 5.1875}, and a group of C+Di corresponding to a constellation point set of the first quadrant has 256 elements such as {1+i, 1+1.020833333×i, 1.020833333+5.1875+5.1875×i}, the complete set of 1D NU 4096-QAM constellation points may be obtained by indicating an arbitrary element a in the group, as a*, −a* and −a. Here, * indicates complex conjugation.

As described above, in the 1D-NUC of Tables 5 and 6, the constellation can be described by the levels at which the constellations occur in the real positive part. The constellation points can be deduced by using the real/imaginary symmetry but also the symmetry of the four quadrants.

Meanwhile, the inventive concept is not limited to the constellations defined in Tables 2-6.

For example, when different sizes of normalization and rounding-off are applied for the values of constellation points defined in Tables 2-6, the values can be indicated as in Tables 7-11. In this case, the constellation defined in Tables 7-11 can be an exemplary embodiment.

Tables 7-11 illustrate the set of constellation points for one quadrant only, but it is obvious to obtain a complete set of constellation points by indicating the constellation point a in the one quadrant, as a*, −a* and −a. Here, * indicates complex conjugation.

To be specific, Table 7 indicates the values of constellation points of 2D NU 16-QAM constellation (2D 16NUC) which is obtained by applying normalization and rounding-off of the values of constellation points defined in Table 2.

TABLE 7 w/Shape NUC_16_2/15 NUC_16_3/15 NUC_16_4/15 w0 0.7073 + 0.7075i 0.7098 + 0.7102i 1.1070 + 0.5806i w1 0.7073 + 0.7074i 0.7050 + 0.7101i 0.5814 + 1.1070i w2 0.7060 + 0.7077i 0.7093 + 0.7039i 0.5525 + 0.3626i w3 0.7065 + 0.7071i 0.7045 + 0.7040i 0.3629 + 0.5527i

In this case, the values of constellation points of other quadrants can be determined by symmetry.

Table 8 indicates values of constellation points of a 2D NU 64-QAM constellation (2D 64NUC) which is obtained by normalization and applying rounding-off of the values of constellation points defined in Table 3.

TABLE 8 w/Shape NUC_64_2/15 NUC_64_3/15 NUC_64_4/15 w0 0.6474 + 0.9831i 0.5472 + 1.1591i 0.5008 + 1.2136i w1 0.6438 + 0.9829i 0.5473 + 1.1573i 0.4994 + 1.2194i w2 0.6471 + 0.9767i 0.5467 + 1.1599i 0.5313 + 1.1715i w3 0.6444 + 0.9762i 0.5479 + 1.1585i 0.5299 + 1.1788i w4 0.9839 + 0.6475i 1.1578 + 0.5478i 1.2107 + 0.5037i w5 0.9778 + 0.6474i 1.1576 + 0.5475i 1.2209 + 0.5008i w6 0.9835 + 0.6434i 1.1591 + 0.5475i 1.1715 + 0.5299i w7 0.9777 + 0.6433i 1.1591 + 0.5475i 1.1802 + 0.5270i w8 0.4659 + 0.6393i 0.3163 + 0.5072i 0.2744 + 0.4762i w9 0.4643 + 0.6386i 0.3163 + 0.5072i 0.2729 + 0.4762i  w10 0.4661 + 0.6353i 0.3163 + 0.5072i 0.2773 + 0.4791i  w11 0.4639 + 0.6350i 0.3163 + 0.5072i 0.2773 + 0.4791i  w12 0.6378 + 0.4671i 0.5087 + 0.3163i 0.4762 + 0.2729i  w13 0.6352 + 0.4673i 0.5087 + 0.3163i 0.4762 + 0.2729i  w14 0.6385 + 0.4656i 0.5087 + 0.3163i 0.4791 + 0.2773i  w15 0.6353 + 0.4653i 0.5087 + 0.3163i 0.4791 + 0.2758i

In this case, the values of constellation points of other quadrants can be determined by symmetry.

Table 9 indicates values of constellation points of a 2D NU 256-QAM constellation (2D 256NUC) which is obtained by applying normalization and rounding-off of the values of constellation points defined in Table 4.

TABLE 9 w/Shape NUC_256_2/15 NUC_256_3/15 NUC_256_4/15 w0  0.5553 + 1.1262i 0.5229 + 1.1810i 0.2975 + 1.0564i w1  0.5673 + 1.1336i 0.5384 + 1.1625i 0.5862 + 0.9617i w2  0.5593 + 1.1204i 0.5148 + 1.1943i 0.2909 + 1.0696i w3  0.5636 + 1.1321i 0.5288 + 1.1751i 0.5796 + 0.9689i w4  0.5525 + 1.1249i 0.4985 + 1.2202i 0.2953 + 1.3357i w5  0.5637 + 1.1320i 0.5111 + 1.1973i 0.7488 + 1.2365i w6  0.5598 + 1.1181i 0.4889 + 1.2357i 0.3004 + 1.5114i w7  0.5659 + 1.1274i 0.5045 + 1.2113i 0.8151 + 1.3816i w8  0.5579 + 1.1381i 0.5222 + 1.1817i 0.3004 + 1.0535i w9  0.5617 + 1.1471i 0.5370 + 1.1640i 0.5847 + 0.9631i w10 0.5593 + 1.1346i 0.5133 + 1.1950i 0.2931 + 1.0659i w11 0.5672 + 1.1430i 0.5303 + 1.1751i 0.5825 + 0.9668i w12 0.5533 + 1.1355i 0.4971 + 1.2216i 0.2953 + 1.3189i w13 0.5632 + 1.1421i 0.5126 + 1.1995i 0.7466 + 1.2168i w14 0.5567 + 1.1325i 0.4882 + 1.2371i 0.2960 + 1.4654i w15 0.5641 + 1.1363i 0.5045 + 1.2128i 0.8297 + 1.3539i w16 1.1309 + 0.5597i 1.1795 + 0.5251i 1.0637 + 0.2960i w17 1.1405 + 0.5660i 1.1625 + 0.5384i 0.9617 + 0.5811i w18 1.1348 + 0.5588i 1.1914 + 0.5133i 1.0732 + 0.2931i w19 1.1491 + 0.5638i 1.1744 + 0.5296i 0.9682 + 0.5818i w20 1.1245 + 0.5615i 1.2209 + 0.4993i 1.3619 + 0.2997i w21 1.1333 + 0.5627i 1.2002 + 0.5148i 1.2249 + 0.7546i w22 1.1284 + 0.5578i 1.2342 + 0.4882i 1.5427 + 0.3106i w23 1.1436 + 0.5636i 1.2142 + 0.5052i 1.3969 + 0.8523i w24 1.1196 + 0.5620i 1.1803 + 0.5229i 1.0615 + 0.2945i w25 1.1347 + 0.5665i 1.1640 + 0.5399i 0.9631 + 0.5818i w26 1.1379 + 0.5611i 1.1921 + 0.5133i 1.0710 + 0.2924i w27 1.1440 + 0.5638i 1.1758 + 0.5303i 0.9675 + 0.5825i w28 1.1221 + 0.5594i 1.2209 + 0.4971i 1.3255 + 0.2975i w29 1.1318 + 0.5686i 1.2024 + 0.5148i 1.1979 + 0.7495i w30 1.1302 + 0.5619i 1.2349 + 0.4889i 1.4560 + 0.3040i w31 1.1386 + 0.5662i 1.2150 + 0.5045i 1.3269 + 0.8414i w32 0.3394 + 0.5381i 0.2740 + 0.4771i 0.2493 + 0.5585i w33 0.3397 + 0.5360i 0.2762 + 0.4801i 0.2960 + 0.5344i w34 0.3387 + 0.5324i 0.2733 + 0.4757i 0.2450 + 0.5417i w35 0.3400 + 0.5335i 0.2748 + 0.4779i 0.2873 + 0.5191i w36 0.3374 + 0.5306i 0.2703 + 0.4742i 0.2049 + 0.3922i w37 0.3405 + 0.5343i 0.2725 + 0.4764i 0.2173 + 0.3806i w38 0.3379 + 0.5324i 0.2696 + 0.4727i 0.1990 + 0.3755i w39 0.3400 + 0.5317i 0.2718 + 0.4749i 0.2107 + 0.3645i w40 0.3397 + 0.5370i 0.2740 + 0.4779i 0.2493 + 0.5599i w41 0.3400 + 0.5383i 0.2755 + 0.4793i 0.2975 + 0.5351i w42 0.3381 + 0.5347i 0.2725 + 0.4757i 0.2450 + 0.5439i w43 0.3382 + 0.5347i 0.2748 + 0.4779i 0.2887 + 0.5213i w44 0.3379 + 0.5342i 0.2711 + 0.4734i 0.2056 + 0.3937i w45 0.3389 + 0.5332i 0.2725 + 0.4764i 0.2187 + 0.3820i w46 0.3402 + 0.5347i 0.2696 + 0.4720i 0.1998 + 0.3762i w47 0.3384 + 0.5340i 0.2711 + 0.4742i 0.2122 + 0.3667i w48 0.5350 + 0.3394i 0.4771 + 0.2740i 0.5607 + 0.2486i w49 0.5363 + 0.3397i 0.4786 + 0.2762i 0.5381 + 0.2960i w50 0.5342 + 0.3389i 0.4764 + 0.2725i 0.5439 + 0.2442i w51 0.5384 + 0.3380i 0.4771 + 0.2748i 0.5220 + 0.2865i w52 0.5329 + 0.3363i 0.4734 + 0.2703i 0.3908 + 0.2049i w53 0.5330 + 0.3387i 0.4757 + 0.2725i 0.3813 + 0.2173i w54 0.5311 + 0.3389i 0.4734 + 0.2696i 0.3740 + 0.1998i w55 0.5332 + 0.3380i 0.4742 + 0.2711i 0.3653 + 0.2100i w56 0.5313 + 0.3397i 0.4771 + 0.2740i 0.5643 + 0.2486i w57 0.5324 + 0.3400i 0.4779 + 0.2762i 0.5410 + 0.2967i w58 0.5339 + 0.3402i 0.4764 + 0.2725i 0.5475 + 0.2435i w59 0.5360 + 0.3405i 0.4771 + 0.2748i 0.5257 + 0.2880i w60 0.5285 + 0.3397i 0.4742 + 0.2703i 0.3937 + 0.2049i w61 0.5317 + 0.3379i 0.4749 + 0.2725i 0.3850 + 0.2187i w62 0.5319 + 0.3381i 0.4734 + 0.2696i 0.3762 + 0.1998i w63 0.5327 + 0.3395i 0.4749 + 0.2711i 0.3689 + 0.2114i

In this case, the values of constellation points of other quadrants can be determined by symmetry.

Table 10 indicates values of constellation points of a 1D NU 1024-QAM constellation (1D 1024NUC) which is obtained by applying normalization and rounding-off of the values of constellation points defined in Table 5.

TABLE 10 u/Shape NUC_1k_2/15 NUC_1k_3/15 NUC_1k_4/15 u0 0.3317 0.2382 0.1924 u1 0.3321 0.2556 0.1940 u2 0.3322 0.2749 0.2070 u3 0.3321 0.2558 0.2050 u4 0.3327 0.2748 0.3056 u5 0.3328 0.2949 0.3096 u6 0.3322 0.2749 0.2890 u7 0.3322 0.2558 0.2854 u8 0.9369 0.9486 0.7167 u9 0.9418 0.8348 0.7362  u10 0.9514 0.7810 0.7500  u11 0.9471 0.8348 0.7326  u12 0.9448 0.9463 0.9667  u13 0.9492 0.8336 0.9665  u14 0.9394 0.9459 1.1332  u15 0.9349 1.4299 1.4761

Table 11 indicates values of constellation points of a 1D NU 4096-QAM constellation (1D 4096NUC) which is obtained by applying normalization and rounding-off of the values of constellation points defined in Table 6.

TABLE 11 u/Shape NUC_4k_2/15 NUC_4k_3/15 NUC_4k_4/15 NUC_4k_5/15 u0 0.2826 0.2038 0.1508 0.1257 u1 0.2885 0.2038 0.1468 0.1257 u2 0.2944 0.2155 0.1456 0.1257 u3 0.2885 0.2155 0.1479 0.1257 u4 0.2944 0.2155 0.1491 0.1257 u5 0.3003 0.2155 0.1444 0.1257 u6 0.2944 0.2097 0.1491 0.1257 u7 0.2885 0.2038 0.1508 0.1257 u8 0.2944 0.2796 0.3368 0.3599 u9 0.3003 0.2912 0.3368 0.3599  u10 0.3003 0.3029 0.3334 0.3484  u11 0.3003 0.2970 0.3363 0.3484  u12 0.2944 0.2970 0.3386 0.3484  u13 0.3003 0.3029 0.3357 0.3484  u14 0.2944 0.2796 0.3340 0.3599  u15 0.2885 0.2796 0.3374 0.3599  u16 0.9714 0.7222 0.6448 0.6112  u17 0.8596 0.7397 0.6569 0.6112  u18 0.7889 0.7455 0.7101 0.6969  u19 0.8478 0.7339 0.6979 0.7026  u20 0.8242 0.7397 0.6974 0.6969  u21 0.7771 0.7513 0.7124 0.6969  u22 0.8360 0.7455 0.6575 0.6112  u23 0.9184 0.7339 0.6465 0.6112  u24 1.1657 1.3046 1.3549 1.4052  u25 0.9479 1.0833 1.1931 1.2281  u26 0.8419 0.9785 1.0117 1.0054  u27 0.9302 1.0134 0.9857 0.9482  u28 0.9950 0.9901 0.9689 0.9425  u29 0.8713 0.9610 0.9967 0.9939  u30 1.0185 1.0658 1.1683 1.1882  u31 1.4660 1.6424 1.6391 1.6566

Meanwhile, it needs to be noted that the method for obtaining complete set of constellation points as in Tables 10 and 11 is completely the same as the method described in Tables 5 and 6.

Meanwhile, those skilled in the art may recognize that rotation, scaling (here, the scaling factor applied to a real axis and an imaginary axis can be the same or different) or other transformation can be applied with respect to the constellation described above. The constellation indicates a comparative position of constellation points, and other constellation can be deduced through rotation, scaling, or other transformation.

In addition, those skilled in the art can recognize that the inventive concept is not limited to constellation defined in Tables 2-11 described above.

For example, in certain exemplary embodiments, a constellation having different order and/or a constellation including a different arrangement or a comparative position of constellation points can be used. As another example, a constellation which is similar to one of constellations defined in Tables 2-11 can be used.

For example, a constellation which has values of constellation points with differences which do not exceed a predetermined threshold (or error) from the values indicated in Tables 2-11 can be used. Here, the threshold value can be expressed as comparative numbers (for example, 0.1%, 1%, 5%, etc.), absolute numbers (for example, 0.001, 0.01, 0.1, etc.) or appropriate methods (rounding-off, flooring, ceiling, or the like). As an example of rounding-off, constellation point “0.707316+0.707473i” can be approximated to “0.7073+0.7075i” by rounding-off at the five decimal places.

In addition, a transmitter and a receiver may use different constellations. For example, a transmitter and a receiver may use respective constellations which have at least one constellation point that has a difference which does not exceed a predetermined threshold value. For example, a receiver may use a constellation having at least one round off/round down constellation point (for example, A2) to demap constellation points, whereas a transmitter may use a constellation having non-round off/round-down constellation points (for example, A1).

In addition, even if an order of the values in Tables 2-11 is changed, the set of constellation points itself is not changed, and thus, it is possible to arrange the values by changing the order of values as in Tables 2-11.

Hereinbelow, an example of a normalization method and an exemplary embodiment of constituting 2D constellation from a 1D level set will be described.

For example, in Table 12, it is assumed that values of constellation points of a 1D NU 1K QAM constellation for a 13/15 coding rate are as shown below.

TABLE 121 Coding Rate Level 13/15 1 1 2 2.975413 3 4.997551 4 7.018692 5 9.102872 6 11.22209 7 13.42392 8 15.69921 9 18.09371 10 20.61366 11 23.2898 12 26.15568 13 29.23992 14 32.59361 15 36.30895 16 40.58404

Here, when a level vector A is indicated as A=(a_(i)), (I=0, 1, 2, . . . , L−1), first of all, the vector A is normalized using Equation 3 shown below, and normalized vector A can be obtained.

$\begin{matrix} {\overset{\_}{A} = \frac{A}{\sqrt{\frac{2}{L}{\sum_{i}a_{i}^{2}}}}} & \left\lbrack {{Equation}3} \right\rbrack \end{matrix}$

In above Equation 3, L indicates the number of level (that is, dimensionality of A). For example, in the case of 16-QAM, 64-QAM, 256-QAM, 1024-QAM, and 4096-QAM, dimensionality of level can be 4, 6, 8, 10 and 12 respectively.

In the example described above, the normalized vector A can be indicated as Table 13 shown below.

TABLE 13 Coding Rate Level 13/15 1 0.0325 2 0.0967 3 0.1623 4 0.228 5 0.2957 6 0.3645 7 0.4361 8 0.51 9 0.5878 10 0.6696 11 0.7566 12 0.8497 13 0.9498 14 1.0588 15 1.1795 16 1.3184

If the normalization method described above is applied to Tables 5 and 6 respectively, it can be easily recognized that Tables 10 and 11 will be obtained respectively.

Next, a final constellation is generated such that all the possible combinations of the real part and the imaginary part, which are the same as one of the entries (that is, components). In this case, for an example, gray mapping can be used.

In the example described above, constellation points in the final first quadrant can be indicated as in Table 14 shown below.

TABLE 14 Label (int.) Constellation Point 1 1.3184 + 1.3184i 2 1.3184 + 1.1795i 3 1.1795 + 1.3184i 4 1.1795 + 1.1795i 5 1.3184 + 0.9498i 6 1.3184 + 1.0588i 7 1.1795 + 0.9498i 8 1.1795 + 1.0588i 9 0.9498 + 1.3184i 10 0.9498 + 1.1795i 11 1.0588 + 1.3184i 12 1.0588 + 1.1795i 13 0.9498 + 0.9498i 14 0.9498 + 1.0588i 15 1.0588 + 0.9498i 16 1.0588 + 1.0588i 17 1.3184 + 0.5878i 18 1.3184 + 0.6696i 19 1.1795 + 0.5878i 20 1.1795 + 0.6696i 21 1.3184 + 0.8497i 22 1.3184 + 0.7566i 23 1.1795 + 0.8497i 24 1.1795 + 0.7566i 25 0.9498 + 0.5878i 26 0.9498 + 0.6696i 27 1.0588 + 0.5878i 28 1.0588 + 0.6696i 29 0.9498 + 0.8497i 30 0.9498 + 0.7566i 31 1.0588 + 0.8497i 32 1.0588 + 0.7566i 33 0.5878 + 1.1795i 34 0.6696 + 1.3184i 35 0.6696 + 1.1795i 36 0.5878 + 0.9498i 37 0.5878 + 1.0588i 38 0.6696 + 0.9498i 39 0.6696 + 1.0588i 40 0.8497 + 1.3184i 41 0.8497 + 1.1795i 42 0.7566 + 1.3184i 43 0.7566 + 1.1795i 44 0.8497 + 0.9498i 45 0.8497 + 1.0588i 46 0.7566 + 0.9498i 47 0.7566 + 1.0588i 48 0.5878 + 0.5878i 49 0.5878 + 0.6696i 50 0.6696 + 0.5878i 51 0.6696 + 0.6696i 52 0.5878 + 0.8497i 53 0.5878 + 0.7566i 54 0.6696 + 0.8497i 55 0.6696 + 0.7566i 56 0.8497 + 0.5878i 57 0.8497 + 0.6696i 58 0.7566 + 0.5878i 59 0.7566 + 0.6696i 60 0.8497 + 0.8497i 61 0.8497 + 0.7566i 62 0.7566 + 0.8497i 63 0.7566 + 0.7566i 64 1.3184 + 0.0325i 65 1.3184 + 0.0967i 66 1.1795 + 0.0325i 67 1.1795 + 0.0967i 68 1.3184 + 0.2280i 69 1.3184 + 0.1623i 70 1.1795 + 0.2280i 71 1.1795 + 0.1623i 72 0.9498 + 0.0325i 73 0.9498 + 0.0967i 74 1.0588 + 0.0325i 75 1.0588 + 0.0967i 76 0.9498 + 0.2280i 77 0.9498 + 0.1623i 78 1.0588 + 0.2280i 79 1.0588 + 0.1623i 80 1.3184 + 0.5100i 81 1.3184 + 0.4361i 82 1.1795 + 0.5100i 83 1.1795 + 0.4361i 84 1.3184 + 0.2957i 85 1.3184 + 0.3645i 86 1.1795 + 0.2957i 87 1.1795 + 0.3645i 88 0.9498 + 0.5100i 89 0.9498 + 0.4361i 90 1.0588 + 0.5100i 91 1.0588 + 0.4361i 92 0.9498 + 0.2957i 93 0.9498 + 0.3645i 94 1.0588 + 0.2957i 95 1.0588 + 0.3645i 96 0.5878 + 0.0325i 97 0.5878 + 0.0967i 98 0.6696 + 0.0325i 99 0.6696 + 0.0967i 100 0.5878 + 0.2280i 101 0.5878 + 0.1623i 102 0.6696 + 0.2280i 103 0.6696 + 0.1623i 104 0.8497 + 0.0325i 105 0.8497 + 0.0967i 106 0.7566 + 0.0325i 107 0.7566 + 0.0967i 108 0.8497 + 0.2280i 109 0.8497 + 0.1623i 110 0.7566 + 0.2280i 111 0.7566 + 0.1623i 112 0.5878 + 0.5100i 113 0.5878 + 0.4361i 114 0.6696 + 0.5100i 115 0.6696 + 0.4361i 116 0.5878 + 0.2957i 117 0.5878 + 0.3645i 118 0.6696 + 0.2957i 119 0.6696 + 0.3645i 120 0.8497 + 0.5100i 121 0.8497 + 0.4361i 122 0.7566 + 0.5100i 123 0.7566 + 0.4361i 124 0.8497 + 0.2957i 125 0.8497 + 0.3645i 126 0.7566 + 0.2957i 127 0.7566 + 0.3645i 128 0.0325 + 1.3184i 129 0.0325 + 1.1795i 130 0.0967 + 1.3184i 131 0.0967 + 1.1795i 132 0.0325 + 0.9498i 133 0.0325 + 1.0588i 134 0.0967 + 0.9498i 135 0.0967 + 1.0588i 136 0.2280 + 1.3184i 137 0.2280 + 1.1795i 138 0.1623 + 1.3184i 139 0.1623 + 1.1795i 140 0.2280 + 0.9498i 141 0.2280 + 1.0588i 142 0.1623 + 0.9498i 143 0.1623 + 1.0588i 144 0.0325 + 0.5878i 145 0.0325 + 0.6696i 146 0.0967 + 0.5878i 147 0.0967 + 0.6696i 148 0.0325 + 0.8497i 149 0.0325 + 0.7566i 150 0.0967 + 0.8497i 151 0.0967 + 0.7566i 152 0.2280 + 0.5878i 153 0.2280 + 0.6696i 154 0.1623 + 0.5878i 155 0.1623 + 0.6696i 156 0.2280 + 0.8497i 157 0.2280 + 0.7566i 158 0.1623 + 0.8497i 159 0.1623 + 0.7566i 160 0.5100 + 1.3184i 161 0.5100 + 1.1795i 162 0.4361 + 1.3184i 163 0.4361 + 1.1795i 164 0.5100 + 0.9498i 165 0.5100 + 1.0588i 166 0.4361 + 0.9498i 167 0.4361 + 1.0588i 168 0.2957 + 1.3184i 169 0.2957 + 1.1795i 170 0.3645 + 1.3184i 171 0.3645 + 1.1795i 172 0.2957 + 0.9498i 173 0.2957 + 1.0588i 174 0.3645 + 0.9498i 175 0.3645 + 1.0588i 176 0.5100 + 0.5878i 177 0.5100 + 0.6696i 178 0.4361 + 0.5878i 179 0.4361 + 0.6696i 180 0.5100 + 0.8497i 181 0.5100 + 0.7566i 182 0.4361 + 0.8497i 183 0.4361 + 0.7566i 184 0.2957 + 0.5878i 185 0.2957 + 0.6696i 186 0.3645 + 0.5878i 187 0.3645 + 0.6696i 188 0.2957 + 0.8497i 189 0.2957 + 0.7566i 190 0.3645 + 0.8497i 191 0.3645 + 0.7566i 192 0.0325 + 0.0325i 193 0.0325 + 0.0967i 194 0.0967 + 0.0325i 195 0.0967 + 0.0967i 196 0.0325 + 0.2280i 197 0.0325 + 0.1623i 198 0.0967 + 0.2280i 199 0.0967 + 0.1623i 200 0.2280 + 0.0325i 201 0.2280 + 0.0967i 202 0.1623 + 0.0325i 203 0.1623 + 0.0967i 204 0.2280 + 0.2280i 205 0.2280 + 0.1623i 206 0.1623 + 0.2280i 207 0.1623 + 0.1623i 208 0.0325 + 0.5100i 209 0.0325 + 0.4361i 210 0.0967 + 0.5100i 211 0.0967 + 0.4361i 212 0.0325 + 0.2957i 213 0.0325 + 0.3645i 214 0.0967 + 0.2957i 215 0.0967 + 0.3645i 216 0.2280 + 0.5100i 217 0.2280 + 0.4361i 218 0.1623 + 0.5100i 219 0.1623 + 0.4361i 220 0.2280 + 0.2957i 221 0.2280 + 0.3645i 222 0.1623 + 0.2957i 223 0.1623 + 0.3645i 224 0.5100 + 0.0325i 225 0.5100 + 0.0967i 226 0.4361 + 0.0325i 227 0.4361 + 0.0967i 228 0.5100 + 0.2280i 229 0.5100 + 0.1623i 230 0.4361 + 0.2280i 231 0.4361 + 0.1623i 232 0.2957 + 0.0325i 233 0.2957 + 0.0967i 234 0.3645 + 0.0325i 235 0.3645 + 0.0967i 236 0.2957 + 0.2280i 237 0.2957 + 0.1623i 238 0.3645 + 0.2280i 239 0.3645 + 0.1623i 240 0.5100 + 0.5100i 241 0.5100 + 0.4361i 242 0.4361 + 0.5100i 243 0.4361 + 0.4361i 244 0.5100 + 0.2957i 245 0.5100 + 0.3645i 246 0.4361 + 0.2957i 247 0.4361 + 0.3645i 248 0.2957 + 0.5100i 249 0.2957 + 0.4361i 250 0.3645 + 0.5100i 251 0.3645 + 0.4361i 252 0.2957 + 0.2957i 253 0.2957 + 0.3645i 254 0.3645 + 0.2957i 255 0.3645 + 0.3645i 256 1.3184 − 1.3184i

FIG. 35 is a block diagram to describe a configuration of a transmitting apparatus according to an exemplary embodiment. Referring to FIG. 35 , the transmitting apparatus 3500 includes an encoder 3510, an interleaver 3520, and a modulator 3530 (or, ‘constellation mapper’).

The encoder 3510 performs channel encoding with respect to input bits and generates a codeword.

For example, the encoder 3510 may perform LDPC encoding with respect to the bits and generate an LDPC codeword using an LDPC encoder (not shown).

Specifically, the encoder 3510 may perform LDPC encoding with the input bits as the information word bits, and generate the LDPC codeword constituting the information word bits and parity bits (that is, the LDPC parity bits). In this case, the LDPC code is a systematic code, the information word may be included in the LDPC codeword as it is.

Herein, the LDPC codeword is constituted by the information word bits and parity bits. For example, the LDPC codeword has N_(ldpc) bits, and may include the information word bits formed of K_(ldpc) bits and parity bits formed of N_(parity)=N_(ldpc)−K_(ldpc) parity bits.

In this case, the encoder 3510 may perform LDPC encoding based on a parity check matrix and generate the LDPC codeword. That is, a process of performing LDPC encoding is a process of generating the LDPC codeword satisfying H·C^(T)=0, and thus, the encoder 3510 may use the parity check matrix when performing LDPC encoding. Herein, H is a parity check matrix and C is an LDPC codeword.

To do this, the transmitting apparatus 3500 may include a separate memory and prestore various types of a parity check matrix.

However, this is merely exemplary, and channel encoding may be performed in various schemes.

The encoder 3510 may perform channel encoding using various coding rates such as 2/15, 3/15, 4/15, 5/15, 6/15, 7/15, 8/15, 9/15, 10/15, 11/15, 12/15 and 13/15. In addition, the encoder 3510 may generate a codeword having various lengths such as 16200 and 64800 based on a length of the bits and coding rate.

An interleaver 3520 interleaves the codeword. That is, the interleaver 3520, based on various interleaving rules, may perform bit-interleaving of the codeword generated by the encoder 3510.

A modulator 3530 maps the codeword which is interleaved according to a modulation scheme onto a non-uniform constellation.

Specifically, the modulator 3530 may perform serial-to-parallel conversion with respect to the interleaved codeword, and demultiplex the interleaved codeword into a cell (or a cell word) formed of a certain number of bits.

For example, the modulator 3530 may receive the codeword bits Q=(q₀, q₁, q₂, . . . ) output from the interleaver 3520, and generates cells.

In this case, the number of bits constituting each cell may be the same as the number of bits constituting a modulation symbol (that is, a modulation order). For example, when the modulator 3530 performs modulation using QPSK, 16-QAM, 64-QAM, 256-QAM, 1024-QAM, 4096-QAM, the number of bits η_(MOD) constituting the modulation symbol may be 2, 4, 6, 8, 10 and 12.

For example, when the modulation scheme is 64-QAM, ηMOD is 6 (η_(MOD)=6), and thus, each cell may be composed as (q₀, q₁, q₂, q₃, q₄, q₅), (q₆, q₇, q₈, q₉, q₁₀, q₁₁), (q₁₂, q₁₃, q₁₄, q₁₅, q₁₆, q₁₇), . . . .

In addition, the modulator 3530 may perform modulation by mapping the cells onto the non-uniform constellation.

Specifically, each cell includes bits as many as the number constituting the modulation symbol, and thus, the modulator 3530 may generate the modulation symbol by sequentially mapping each cell onto a constellation point of the non-uniform constellation. Herein, the modulation symbol corresponds to a constellation point of a constellation.

In this case, constellation may include constellation points which are defined based on Tables 2-11 according to a modulation scheme.

To be specific, the constellation may include the constellation points which are defined by a constellation position vector as in Tables 2-4 and 7-9, according to a modulation scheme. Or, the constellation may include the constellation points which are defined by the constellation position vector which is generated based on the level set as in Tables 5, 6, 10, and 11 according to a modulation scheme.

That is, the modulator 3530, in consideration of the coding rate used for encoding by the encoder 3510, may perform modulation by mapping cells onto the set of constellation points which corresponds to the coding rate from among the sets of constellation points which are defined based on Tables 2-11 according to the coding rates.

For example, constellation may include constellation points which are defined based on Table 8, when a modulation scheme is 64-QAM.

To be specific, the modulator 3530, when encoding is performed with the coding rate of 2/15 by the encoder 3510, may map the interleaved codeword onto the non-uniform constellation which includes constellation points defined by NUC_64_2/15 of Table 8.

That is, when the coding rate is 2/15 and modulation is performed to 2D 64NUC, the constellation points in the first quadrant of constellation can be expressed as the constellation position vector {w₀, w₁, w₂, . . . , w₁₄, w₁₅}={0.6474+0.9831i, 0.6438+0.9829i, 0.6471+0.9767i, . . . , 0.6385+0.4656i, 0.6353+0.4653i} which is defined as NUC_64_2/15 of Table 8.

In addition, the modulator 3530, when encoding is performed with the coding rate of 3/15 by the encoder 3510, may map the interleaved codeword onto the non-uniform constellation which includes the constellation points defined by NUC_64_3/15 of Table 8.

That is, when the coding rate is 3/14 and modulation is performed to 2D 64NUC, the constellation points in the first quadrant of constellation can be expressed as the constellation position vector {w₀, w₁, w₂, . . . , w₁₄, w₁₅}={0.5472+1.1591i, 0.5473+1.1573i, 0.5467+1.1599i, . . . , 0.5087+0.3163i, 0.5087+0.3163i} which is defined as NUC_64_3/15 of Table 8.

In addition, the modulator 3530, when encoding is performed with the coding rate of 4/15 by the encoder 3510, may map the interleaved codeword onto the non-uniform constellation which includes the constellation points defined by NUC_64_4/15 of Table 8.

That is, when coding rate is 4/15 and modulation is performed to 2D 64NUC, the constellation points of the first quadrant of constellation may be expressed as the constellation position vector {w₀, w₁, w₂, . . . , w₁₄, w₁₅}={0.5008+1.2136i, 0.4994+1.2194i, 0.5313+1.1715i, . . . , 0.4791+0.2773i, 0.4791+0.2758i} which is defined as NUC_64_4/15 of Table 8.

Table 8 indicates the constellation points in one quadrant of constellation, and the constellation points in remaining quadrants of constellation may be obtained by indicating each constellation point a, which is defined in Table 8, as a*, −a* and −a respectively. (Here, * indicates complex conjugation).

As another example, when the modulation scheme is 256-QAM, the constellation points which are defined based on Table 9 may be included.

Specifically, the modulator 3530, when encoding is performed with the coding rate of 2/15 by the encoder 3510, may map the interleaved codeword onto the non-uniform constellation which includes the constellation points defined by NUC_256_2/15 of Table 9.

That is, when coding rate is 2/15 and modulation is performed to 2D 256NUC, the constellation points of the first quadrant of constellation may be expressed as the constellation position vector {w₀, w₁, w₂, . . . , w₆₂, w₆₃}={0.5553+1.1262i, 0.5673+0.1336i, 0.5593+1.1204i, . . . , 0.5319+0.3381i, 0.5327+0.3395i} which is defined as NUC_256_2/15 of Table 9.

In addition, the modulator 3530, when encoding is performed with the coding rate of 3/15 by the encoder 3510, may map the interleaved codeword onto the non-uniform constellation which includes the constellation points defined by NUC_256_3/15 of Table 9.

That is, when coding rate is 3/15 and modulation is performed to 2D 256NUC, the constellation points of the first quadrant of constellation may be expressed as the constellation position vector {w₀, w₁, w₂, . . . , w₆₂, w₆₃}={0.5229+1.1810i, 0.5384+1.1625i, 0.5148+1.1943i, . . . , 0.4734+0.2696i, 0.4749+0.2711i} which is defined as NUC_256_3/15 of Table 9.

In addition, the modulator 3530, when encoding is performed with the coding rate of 4/15 by the encoder 3510, may map the interleaved codeword onto the non-uniform constellation which includes the constellation points defined by NUC_256_4/15 of Table 8.

That is, when coding rate is 4/15 and modulation is performed to 2D 256NUC, the constellation points of the first quadrant of constellation may be expressed as the constellation position vector {w₀, w₁, w₂, . . . , w₆₂, w₆₃}={0.2975+1.0564i, 0.5862+0.9617i, 0.2909+1.0696i, . . . , 0.3762+0.1998i, 0.3689+0.2114i} which is defined as NUC_256_4/15 of Table 9.

Table 9 indicates the constellation points in one quadrant of constellation, and the constellation points in remaining quadrants of constellation may be obtained by indicating each constellation point a, which is defined in Table 9, as a*, −a* and −a respectively (Here, * indicates complex conjugation).

As another example, constellation, when the modulation scheme is 1024-QAM, may include the constellation points which are defined based on Table 10.

Specifically, the modulator 3530, when encoding is performed with the coding rate of 2/15 by the encoder 3510, may map the interleaved codeword onto the non-uniform constellation which includes the constellation points defined by NUC_1k_2/15 of Table 10.

That is, when coding rate is 2/15 and modulation is performed to 1D 1024NUC, the level set may be A={0.3317, 0.3321, 0.3322, . . . , 0.9394, 0.9349} as NUC_1k_2/15 of Table 10, and the constellation position vector indicating the constellation points in the first quadrant may be expressed as {0.3317+0.3317i, 0.3317+0.3321i, 0.3321+0.3317i, . . . , 0.9349+0.9349i}.

In addition, the modulator 3530, when encoding is performed with the coding rate of 3/15 by the encoder 3510, may map the interleaved codeword onto the non-uniform constellation which includes the constellation points defined by NUC_1k_3/15 of Table 10.

That is, when coding rate is 3/15 and modulation is performed to 1D 1024NUC, level set may be A={0.2382, 0.2556, 0.2749, . . . , 0.9459, 1.4299} as NUC_1k_3/15 of Table 10, and the constellation position vector indicating the constellation points of the first quadrant may be expressed as {0.2382+0.2382i, 0.2382+0.2556i, 0.2556+0.2382i, . . . , 1.4299+1.4299i}.

In addition, the modulator 3530, when encoding is performed with the coding rate of 4/15 by the encoder 3510, may map the interleaved codeword onto the non-uniform constellation which includes the constellation points defined by NUC_1k_4/15 of Table 10.

That is, when coding rate is 4/15 and modulation is performed to 1D 1024NUC, the level set may be A={0.1924, 0.1940, 0.2070, . . . , 1.1332, 1.4761} as NUC_1k_4/15 of Table 10, and the constellation position vector indicating the constellation points in the first quadrant may be expressed as {0.1924+0.1924i, 0.1924+0.1940i, 0.1940+0.1924i, . . . , 1.4761+1.4761i}.

Table 10 is used to define the constellation points in one quadrant of constellation, and the constellation points in remaining quadrants may be obtained by indicating each constellation point, which is defined based on Table 10, as a*, −a* and −a (Here, * indicates complex conjugation).

As another example, constellation, when the modulation scheme is 4096-QAM, may include the constellation points which are defined based on Table 11.

Specifically, the modulator 3530, when encoding is performed with the coding rate 2/15 by the encoder 3510, may map the interleaved codeword onto the non-uniform constellation which includes the constellation points defined by NUC_4k_2/15 of Table 11.

That is, when coding rate is 2/15 and modulation is performed to 1D 4096NUC, the level set may be A={0.2826, 0.2885, 0.2944, . . . , 1.0185, 1.4660} as NUC_4k_2/15 of Table 11, and the constellation position vector indicating the constellation points in the first quadrant may be expressed as {0.2826+0.2826i, 0.2826+0.2885i, 0.2885+0.2826i, . . . , 1.4660+1.4660i}.

In addition, the modulator 3530, when encoding is performed with the coding rate of 3/15 by the encoder 3510, may map the interleaved codeword onto the non-uniform constellation which includes the constellation points defined by NUC_4k_3/15 of Table 11.

That is, when coding rate is 3/15 and modulation is performed to 1D 4096NUC, level set may be A={0.2038, 0.2038, 0.2155, . . . , 1.0658, 1.6424} as NUC_4k_3/15 of Table 11, and the constellation position vector indicating the constellation points in the first quadrant may be expressed as {0.2038+0.2038i, 0.2038+0.2155i, 0.2155+0.2038i, . . . , 1.6424+1.6424i}.

In addition, the modulator 3530, when encoding is performed with the coding rate of 4/15 by the encoder 3510, may map the interleaved codeword onto the non-uniform constellation which includes the constellation points defined by NUC_4k_4/15 of Table 11.

That is, when coding rate is 4/15 and modulation is performed to 1D 4096NUC, the level set may be A={0.1508, 0.1468, 0.1456, . . . , 1.1683, 1.6391} as NUC_4k_4/15 of Table 11, and the constellation position vector indicating the constellation points in the first quadrant may be expressed as {0.1508+0.1508i, 0.1508+0.1468i, 0.1468+0.1508i, . . . , 1.6391+1.6391i}.

In addition, the modulator 3530, when encoding is performed with the coding rate of 4/15 by the encoder 3510, may map the interleaved codeword onto the non-uniform constellation which includes the constellation points defined by NUC_4k_5/15 of Table 11.

That is, when coding rate is 5/15 and modulation is performed to 1D 4096NUC, the level set may be A={0.1257, 0.1257, 0.1257, . . . , 1.1882, 1.6566} as NUC_4k_5/15 of Table 11, and the constellation position vector indicating the constellation points in the first quadrant may be expressed as {0.1257+0.1257i, 0.1257+0.3599i, 0.3599+0.1257i, . . . , 1.6566+1.6566i}.

Table 11 is used to define the constellation points in one quadrant, and the constellation points with respect to the remaining quadrants of constellation may be obtained by indicating each constellation point a, which is defined based on Table 11, as a*, −a* and −a (Here, * indicates complex conjugation).

In the above-described examples, it is described that the cells are mapped onto the set of constellation points which correspond to coding rate used for encoding, but this is merely exemplary, and in some cases, the modulator 3530 may map the cells onto the set of constellation points which do not correspond to coding rate which is used for encoding.

As an example, when 64-QAM is used, even if encoding is performed with the coding rate of 2/15, the modulator 3530 may map the cells onto the set of constellation points which are defined as NUC_64_3/15 or NUC_64_4/15 of Table 8, instead of the set of constellation points which are defined as NUC_64_2/15 of Table 8.

The transmitting apparatus 3500 may modulate a signal which is mapped onto the constellation and transmit the signal to a receiving apparatus (for example, 3600 of FIG. 36 ). For example the transmitting apparatus 3500 may map the signal which is mapped to the constellation onto an orthogonal frequency division multiplexing (OFDM) frame by using an OFDM scheme, and may transmit the signal to the receiving apparatus 3600 via an allocated channel.

In other word, in mapping method for 16-QAM, 64-QAM and 256-QAM, each input data cell word (y_(0,s), . . . , y_(ηMOD−1,s)) shall be modulated using a 2D non-uniform constellation to give a constellation point z_(s). Index s denotes the discrete time index, η_(MOD)=log₂(M), M being the number of constellation points, e.g., M=64 for 64-QAM. The vector of complex constellation points x=(x₀, . . . , x_(M−1)) includes all M constellation points of the QAM alphabet. The k-th element of this vector, x_(k), corresponds to the QAM constellation point for the input cell word (y_(0,s), . . . , y_(ηMOD−1,s)), if these bits take on the decimal number k (y_(0,s) being the most significant bit (MSB), and y_(ηMOD−1,s) being the least significant bit (LSB)). Due to the quadrant symmetry, the complete vector x can be derived by defining just the first quarter of the complex constellation points, i.e., (x₀, . . . , x_(M/4−1)), which corresponds to the first quadrant. The generation rule for the remaining points is described below. Defining b=M/4, the first quarter of complex constellation points is denoted as the NUC position vector w=(w₀, . . . , w_(b−1)). The position vectors are defined the above tables. As an example, the NUC position vector for a 16-QAM comprises the complex constellation points with the labels corresponding to the decimal values 0, i.e., (y_(0,s), . . . , y_(ηMOD−1,s))=0000, to b−1, i.e., (y_(0,s), . . . , y_(ηMOD−1,s))=0011. The remaining constellation points are derived as follows:

(x₀, . . . , x_(b−1)) = w (first quarter) (x_(b), . . . , x_(2b−1)) = −conj(w) (second quarter) (x_(2b), . . . , x_(3b−1)) = conj(w) (third quarter) (x_(3b), . . . , x_(4b−1)) = −w (fourth quarter), with conj being the complex conjugate.

As an example, the NUC position vector for 16-QAM and code rate 2/15 is constructed as follows. From Table 7, w=(0.7073+0.7075i, 0.7073+0.7074i, 0.7060+0.7077i, 0.7065+0.7071i). Here and in the following, i=√(−1) is the imaginary unit. Assuming the input data cell word is (y_(0,s), . . . , y_(ηMOD−1,s))=(1100), the corresponding QAM constellation point at time index s is z_(s)=x₁₂=−w₀=−0.7073-0.7075i.

Also, in mapping method for 1024-QAM and 4096-QAM, Each input data cell word (y_(0,s), . . . , y_(ηMOD−1,s)) at discrete time index s shall be modulated using a 1-dimensional non-uniform QAM constellation to give a constellation point z_(s) prior to normalization. 1-dimensional refers to the fact that a 2-dimensional QAM constellation can be separated into two 1-dimensional PAM constellations, one for each I and Q component. The exact values of the real and imaginary components Re(z_(s)) and Im(z_(s)) for each combination of the relevant input cell word (y_(0,s), . . . , y_(ηMOD−1,s)) are given by a 1D-NUC position vector u=(u₀, . . . , u_(v)), which defines the constellation point positions of the non-uniform constellation in one dimension. The number of elements of the 1D-NUC position vector u is defined by

${v = \frac{\sqrt{M}}{2}}.$

As an example the 1024-NUC for code rate 2/15 is defined by the NUC position vector NUC_1k_2/15. From Table 10, u=(u₀, . . . , u₁₅)=(0.3317, 0.3321, 0.3322, 0.3321, 0.3327, 0.3328, 0.3322, 0.3322, 0.9369, 0.9418, 0.9514, 0.9471, 0.9448, 0.9492, 0.9394, 0.9349). Assuming the input data cell (y_(0,s), . . . , y_(ηMOD−1,s))=(0010011100) the corresponding QAM constellation point z_(s) has Re(z_(s))=u₃=0.3321 (defined by even index bit labels, i.e., 01010) and Im(z_(s))=u₁₁=0.9471 (defined by odd index bit label, i.e., 00110).

FIG. 36 is a block diagram to describe a configuration of the receiving apparatus according to an exemplary embodiment. Referring to FIG. FIG. 36 , the receiving apparatus 3600 includes a demodulator 3610, a deinterleaver 3620, and a decoder 3630.

The demodulator 3610 receives and demodulates a signal transmitted from the transmitting apparatus 3500. Specifically, the demodulator 3610 may generate a value corresponding to the codeword by demodulating the received signal.

In this case, the demodulator 3610 may perform demodulation to correspond to the modulation scheme which is used by the transmitting apparatus 3500. To do this, the transmitting apparatus 3500 may transmit information on the modulation scheme to the receiving apparatus 3600, or the transmitting apparatus 3500 may perform modulation using the modulation scheme which is predefined between the transmitting apparatus 3500 and the receiving apparatus 3600.

Meanwhile, a value which corresponds to the codeword may be expressed as a channel value with respect to the received signal. There may be various methods for determining the channel value, for example, a method for determining a log likelihood ratio (LLR) value is an example of the method for determining the channel value.

The LLR value may indicate a log value for a ratio of the probability that the bit transmitted from the transmitting apparatus 3500 is 0 and the probability that the bit is 1. In addition, the LLR value may be a bit value which is determined by a hard decision, or may be a representative value which is determined according to a section to which the probability that the bit transmitted from the transmitting apparatus 3500 is 0 or 1 belongs.

The demodulator 3610 may perform cell-to-bit conversion with respect to a value corresponding to the codeword and output an LLR value in the unit of bits.

The deinterleaver 3620 deinterleaves an output value of the demodulator 3610, and outputs the value to the decoder 3630.

To be specific, the deinterleaver 3620 is an element corresponding to the interleaver 3520 of the transmitting apparatus 3500 and performs an operation corresponding to the interleaver 3520. That is, the interleaver 3620 performs the interleaving operation of the interleaver 3520 inversely and deinterleaves an LLR value.

The decoder 3630 may perform channel decoding based on the output value of the deinterleaver 3620.

Specifically, the decoder 3630 is an element corresponding to the encoder 3510 of the transmitting apparatus 3500, which may correct an error by performing decoding by using the LLR value output from the deinterleaver 3620.

For example, the decoder 3630 may include an LDPC decoder (not shown) to perform LDPC decoding.

In this case, the decoder 3630 may perform LDPC decoding using an iterative decoding scheme based on a sum-product algorithm. Herein, the sum-product algorithm refers to an algorithm by which messages (e.g., LLR value) are exchanged through an edge on a bipartite graph of a message passing algorithm, and an output message is calculated from messages input to variable nodes or check nodes, and is updated.

Meanwhile, the decoder 3630 may use a parity check matrix for LDPC decoding.

In this case, the parity check matrix which is used for decoding may have the same structure as the parity check matrix which is used for encoding.

Meanwhile, information on the parity check matrix or information on the code rate used for LDPC decoding may be prestored in the receiving apparatus 3600 or provided by the transmitting apparatus 3500.

The foregoing is merely exemplary, and channel decoding may be performed by various schemes which correspond to the channel coding which is performed by the transmitting apparatus 3500.

A non-transitory computer readable medium may be provided which stores a program to operate the above-described methods of various exemplary embodiments, according to an exemplary embodiment. The non-transitory recordable medium refers to a medium which may store data semi-permanently rather than storing data for a short time such as a register, a cache, and a memory and may be readable by an apparatus. Specifically, this medium may be a non-temporal recordable medium such as compact disk (CD), digital versatile disk (DVD), hard disk, Blu-ray disk, universal serial bus (USB), memory card, or read-only memory (ROM), not being limited thereto.

The components, elements, modules or units represented by a block as illustrated in FIGS. 18, 35 and 36 may be embodied as various numbers of hardware, software and/or firmware structures that execute respective functions described above, according to an exemplary embodiment. For example, these components, elements, modules or units may use a direct circuit structure, such as a memory, processing, logic, a look-up table, etc. that may execute the respective functions through controls of one or more microprocessors or other control apparatuses. Also, these components, elements, modules or units may be specifically embodied by a program or a part of code, which contains one or more executable instructions for performing specified logic functions. Also, at least one of these components, elements, modules or units may further include a processor such as a central processing unit (CPU) that performs the respective functions, a microprocessor, or the like. A bus is not illustrated in the above block diagrams of FIGS. 18, 35 and 36 . However, communications between the respective components, elements, modules or units may be carried out through the bus.

The foregoing embodiments and advantages are merely exemplary and should not be construed as limiting the inventive concept. Also, the description of the exemplary embodiments of the inventive concept is intended to be illustrative, and not to limit the scope of the claims, and many alternatives, modifications, and variations will be apparent to those skilled in the art.

FIG. 37 is a flowchart to describe a method for transmitting of a transmitting apparatus according to an exemplary embodiment.

First of all, a codeword is generated (S3710) by performing channel encoding with respect to the bits, and the codeword is interleaved (S3720).

Thereafter, the interleaved codeword is mapped onto the non-uniform constellation according to a modulation scheme (S3730).

In this case, constellation may include the constellation points which are defined based on Tables 2-11 according to a modulation scheme.

As an example, when the modulation scheme is 64-QAM, constellation may include the constellation points which are defined based on Table 8.

Specifically, when encoding is performed with the coding rate of 2/15, at S3730, an interleaved codeword may be mapped onto the non-uniform constellation which includes the constellation points defined by NUC_64_2/15 of Table 8.

When encoding is performed with the coding rate of 3/15, at S3730, the interleaved codeword may be mapped onto the non-uniform constellation which includes the constellation points which are defined by NUC_64_3/15 of Table 8.

In addition, when encoding is performed with the coding rate of 4/15, at S3730, the interleaved codeword may be mapped onto the non-uniform constellation which includes the constellation points defined by NUC_64_4/15 of Table 8.

As another example, constellation may include the constellation points which are defined based on Table 9, when the modulation scheme is 256-QAM.

Specifically, when encoding is performed with the coding rate of 2/15, at S3730, the interleaved codeword may be mapped onto the non-uniform constellation which includes the constellation points defined by NUC_256_2/15 of Table 9.

In addition, when encoding is performed with the coding rate of 3/15, at S3730, the interleaved codeword may be mapped onto the non-uniform constellation which includes the constellation points which are defined by NUC_256_3/15 of Table 9.

When encoding is performed with the coding rate of 4/15, at S3730, the interleaved codeword may be mapped onto the non-uniform constellation which includes the constellation points defined by NUC_256_4/15 of Table 9.

Meanwhile, Tables 9 and 9 indicate the constellation points in one quadrant of the constellation, and the constellation points in the remaining quadrants of constellation may be obtained by indicating each constellation point a, which is defined in Tables 8 and 9, as a*, −a* and −a respectively (Here, * indicates complex conjugation).

As another example, constellation may include, when the modulation scheme is 1024-QAM, the constellation points which are defined based on Table 10.

Specifically, when encoding is performed with the coding rate of 2/15, at S3730, the interleaved codeword may be mapped onto the non-uniform constellation which includes the constellation points defined based on NUC_1k_2/15 of Table 10.

In addition, when encoding is performed with the coding rate of 3/15, at S3730, the interleaved codeword may be mapped onto the non-uniform constellation which includes the constellation points defined based on NUC_1k_3/15 of Table 10.

In addition, when encoding is performed with the coding rate of 4/15, at S3730, the interleaved codeword may be mapped onto the non-uniform constellation which includes the constellation points defined based on NUC_1k_4/15 of Table 10.

As another example, constellation may include the constellation points which are defined based on Table 11, when the modulation scheme is 4096-QAM.

Specifically, when encoding is performed with the coding rate of 2/15, at S3730, the interleaved codeword may be mapped onto the non-uniform constellation which includes the constellation points defined based on NUC_4k_2/15 of Table 11.

In addition, when encoding is performed with the coding rate of 3/15, at S3730, the interleaved codeword may be mapped onto the non-uniform constellation which includes the constellation points defined based on NUC_4k_3/15 of Table 11.

When encoding is performed with the coding rate of 4/15, at S3730, the interleaved codeword may be mapped onto the non-uniform constellation which includes the constellation points which are defined based on NUC_4k_4/15 of Table 11.

When encoding is performed with the coding rate of 5/15, at S3730, the interleaved codeword may be mapped onto the non-uniform constellation which includes the constellation points which are defined based on NUC_4k_5/15 of Table 11.

Meanwhile, Tables 10 and 11 are used to define the constellation points in one quadrant, and the constellations in the remaining quadrants of constellation may be obtained by indicating each constellation point a, which is defined based on Tables 10 and 11, as a*, −a* and −a respectively (Here, * indicates complex conjugation).

In the present disclosure, in order to generate the optimized constellation, capacity needs to be determined. To do this, SNR is an important parameter. However, optimizing with respect to the SNR does not necessarily mean that an environment which satisfies the SNR is necessary. Though it is highly likely that the optimized performance may be obtained in the environment which satisfies the SNR, but in general, receiving SNR may change frequently according to system environment, and it is obvious that different SNR or different channel coding rate may be used according to not only complexity of realizing the system but also various purposes to support several channel environments using the modulation scheme to which one NUC constellation point is applied.

A non-transitory computer readable medium in which a program which sequentially performs non-uniform constellation generation method is stored therein may be provided.

The non-transitory computer-recordable medium is not a medium configured to temporarily store data such as a register, a cache, or a memory but an apparatus-readable medium configured to semi-permanently store data. Specifically, the above-described various applications or programs may be stored in the non-transitory apparatus-readable medium such as a compact disc (CD), a digital versatile disc (DVD), a hard disc, a Blu-ray disc, a universal serial bus (USB), a memory card, or a read only memory (ROM) are provided.

In the block diagram which illustrates the transmitting apparatus and receiving apparatus, bus is nogt illustrated, but communication among elements of each apparatus can be done through the bus. In addition, each apparatus may further include CPU performing the steps described above and processors such as a micro processor.

The foregoing exemplary embodiments and advantages are merely exemplary and are not to be construed as limiting the inventive concept. The exemplary embodiments can be readily applied to other types of device or apparatus. Also, the description of the exemplary embodiments is intended to be illustrative, and not to limit the scope of the inventive concept, and many alternatives, modifications, and variations will be apparent to those skilled in the art. 

What is claimed is:
 1. A receiving method comprising: receiving a signal from a transmitting apparatus; demodulating the signal to generate values; deinterleaving the values; and decoding the deinterleaved values based on a low density parity check (LDPC) code, a code rate of the LDPC code being 3/15, wherein the signal is demodulated based on position vectors w, in which w comprises w₀, w₁, . . . , and w₁₅ and a quadrant of a position vector w_(i) in which i is any one of 0 to 15, and wherein the position vectors w are represented in a table as below: w₀  0.5472 + 1.1591i w₁  0.5473 + 1.1573i w₂  0.5467 + 1.1599i w₃  0.5479 + 1.1585i w₄  1.1578 + 0.5478i w₅  1.1576 + 0.5475i w₆  1.1591 + 0.5475i w₇  1.1591 + 0.5475i w₈  0.3163 + 0.5072i w₉  0.3163 + 0.5072i w₁₀ 0.3163 + 0.5072i w₁₁ 0.3163 + 0.5072i w₁₂ 0.5087 + 0.3163i w₁₃ 0.5087 + 0.3163i w₁₄ 0.5087 + 0.3163i w₁₅ 0.5087 + 0.3163i


2. The receiving method as claimed in claim 1, wherein constellation points for 64-quadrature amplitude modulation (QAM) comprise constellation points in 4 quadrants, wherein the represented position vectors w indicate constellation points in one quadrant, and wherein constellation points in remaining quadrants are obtained by indicating each constellation point a which is defined by the table as a*, −a* and −a respectively, where * indicates complex conjugation.
 3. A transmitting method comprising: interleaving a codeword; and demultiplexing bits of the interleaved codeword to generate cells; and mapping the cells to 2-dimensional (2D) non-uniform constellation points using 64-quadrature amplitude modulation (QAM) and position vectors w, in which w comprises w₀, w₁, . . . , and w₁₅, wherein the codeword comprises input bits and parity bits which are generated by encoding the input bits based on a low density parity check (LDPC) code, a code rate of the LDPC code being 3/15, wherein the cells are mapped to the constellation points by determining a position vector w_(i), in which i is any one of 0 to 15, corresponding to a cell from among the position vectors w, and determining a quadrant of the position vector w_(i), and wherein the position vectors w are represented in a table as below: w₀  0.5472 + 1.1591i w₁  0.5473 + 1.1573i w₂  0.5467 + 1.1599i w₃  0.5479 + 1.1585i w₄  1.1578 + 0.5478i w₅  1.1576 + 0.5475i w₆  1.1591 + 0.5475i w₇  1.1591 + 0.5475i w₈  0.3163 + 0.5072i w₉  0.3163 + 0.5072i w₁₀ 0.3163 + 0.5072i w₁₁ 0.3163 + 0.5072i w₁₂ 0.5087 + 0.3163i w₁₃ 0.5087 + 0.3163i w₁₄ 0.5087 + 0.3163i w₁₅ 0.5087 + 0.3163i


4. The transmitting method as claimed in claim 3, wherein constellation points for 64-QAM comprise constellation points in 4 quadrants, wherein the represented position vectors w indicate constellation points in one quadrant, and wherein constellation points in remaining quadrants are obtained by indicating each constellation point a which is defined by the table as a*, −a* and −a respectively, where * indicates complex conjugation. 